#361304

order	512
nilpotency class	3
isoclinism	isoclinic class of #23860
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

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

1754 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fi,c^2=i,d^2=fi,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,d^b=di,e^b=ei,e^d=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 268500);