#150360

order	512
nilpotency class	5
isoclinism	isoclinic class of #150220
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2127
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	#391	C₂³
prime radical	C₂³	C₂×#99
LCS	G ▷ C₂×C₄² ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂×C₄² ◁ #2327 ◁ G
LECS	G ▷ #391 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

276 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 54 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 72 points:

(2,6,7,8)(9,10,13,19,15,21,31,48)(11,22,24,47,26,39,54,65)(12,17,28,43,27,44,55,70)(14,18,34,46,35,20,32,49)(16,36,38,25,40,30,61,33)(23,50,52,71,51,63,72,68)(29,45,60,69,56,42,58,67)(37,62,64,53,41,57,66,59);(1,2)(3,6)(4,8)(5,7)(9,11,15,26)(10,16,21,40)(12,23,27,51)(13,30,31,36)(14,33,35,25)(17,41,44,37)(18,39,20,22)(19,47,48,65)(24,32,54,34)(28,57,55,62)(29,59,56,53)(38,49,61,46)(42,63,45,50)(43,68,70,71)(52,58,72,60)(64,69,66,67);(1,3,5,4)(2,6,7,8)(9,12)(10,17)(11,23)(13,28)(14,29)(15,27)(16,37)(18,45)(19,43)(20,42)(21,44)(22,50)(24,52)(25,53)(26,51)(30,57)(31,55)(32,58)(33,59)(34,60)(35,56)(36,62)(38,64)(39,63)(40,41)(46,69)(47,71)(48,70)(49,67)(54,72)(61,66)(65,68)

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=gh,d^2=g,e^2=i,f^2=i,b^a=bd,d^a=df,f^a=fh,g^a=gi,h^a=hi,c^b=ch,d^b=dg,e^b=efgi,f^b=fi,d^c=di,e^d=ehi,f^e=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 57556);