#149714

order	512
nilpotency class	5
isoclinism	isoclinic class of #13605
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2127
commutator	C₄²	C₂²×C₈
Frattini, cosocle	#382	C₂³
prime radical	C₂³	C₂×#99
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂×C₄² ◁ #2327 ◁ G
LECS	G ▷ #382 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

291 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 291 conjugacy classes of subgroups — too many to draw. Only its 68 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56910);