#307970

order	512
nilpotency class	3
isoclinism	isoclinic class of #244791
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

607 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=fh,c^2=fi,d^2=h,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,e^b=eh,g^b=gi,g^c=gi,e^d=eh

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 215166);