#372506

order	512
nilpotency class	3
isoclinism	isoclinic class of #24274
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

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	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1508 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=eh,c^2=ei,d^2=fgi,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,d^b=di,f^c=fi,g^c=gi,h^c=hi

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, 279702);