#367238

order	512
nilpotency class	3
isoclinism	isoclinic class of #24378
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×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 ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1102 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 274434);