#228330

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×#121	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₂²⋊C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

649 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=e,f^2=h,g^2=i,b^a=be,c^a=cf,f^a=fh,d^b=deh,d^c=dhi,f^c=fh

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, 135526);