#218571

order	512
nilpotency class	3
isoclinism	isoclinic class of #23924
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

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, prime radical	C₂³	C₂×C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ #1744 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

593 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=h,d^2=eh,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cei,d^b=dh,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, 125767);