#341495

order	512
nilpotency class	3
isoclinism	isoclinic class of #331655
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

561 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 248691);