#225102

order	512
nilpotency class	3
isoclinism	isoclinic class of #23860
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₂²⋊C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

864 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fh,e^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,c^b=ci,d^b=de,e^b=eh,e^d=eh

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, 132298);