#150601

order	512
nilpotency class	5
isoclinism	isoclinic class of #150594
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	C₂⁴	#99
LCS	G ▷ C₂×C₄² ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂³×C₄ ◁ #2310 ◁ G
LECS	G ▷ #391 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

317 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 317 conjugacy classes of subgroups — too many to draw. Only its 52 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=i,d^2=g,e^2=i,f^2=i,b^a=bd,c^a=cg,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^c=ei,e^d=ehi,f^e=fi

Conjugacy classes & character table

32 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 57797);