#325581

order	512
nilpotency class	3
isoclinism	isoclinic class of #278011
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

629 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 232777);