#232051

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

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	C₂×#127	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₄⋊C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

645 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=ef,d^2=eh,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,f^a=fh,c^b=ch,d^b=de,d^c=dhi

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, 139247);