#7191504

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994043
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1419 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

(2,5)(6,8)(9,10,14,11)(12,15,19,17)(13,18,21,16)(20,23,24,22)(25,26,30,34)(27,35,37,40)(28,33,29,32)(31,38,39,36);(1,2)(3,6)(4,5)(7,8)(9,11,14,10)(12,15,19,17)(13,16,21,18)(20,23,24,22);(1,3,4,7)(2,6,5,8)(9,12)(10,15)(11,17)(13,20)(14,19)(16,22)(18,23)(21,24)(25,27,30,37)(26,31,34,39)(28,36,29,38)(32,35,33,40);(2,5)(6,8)(12,19)(15,17)(20,24)(22,23)(25,28,30,29)(26,32,34,33)(27,36,37,38)(31,35,39,40);(9,13,14,21)(10,16,11,18)(12,20,19,24)(15,22,17,23)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=fi,d^2=i,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72282761, #72283173, #72283313, …
maximal normal subgroup of	#72282761, #72283173, #72283313, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7098700);