#365693

order	512
nilpotency class	3
isoclinism	isoclinic class of #23848
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

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

1254 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 272889);