#218280

order	512
nilpotency class	3
isoclinism	isoclinic class of #23924
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	C₂²×D₄	C₂²×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ #1744 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

571 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)(3,7)(9,10,13,15,12,11,14,16)(17,18,21,25)(19,22,26,30)(20,23,28,24)(27,29,32,31);(1,2)(3,6)(4,5)(7,8)(9,11,13,16,12,10,14,15)(17,19)(18,22)(20,27)(21,26)(23,29)(24,31)(25,30)(28,32);(2,5)(6,8)(10,11)(15,16)(18,23)(20,28)(22,29)(24,25)(27,32)(30,31);(1,3,4,7)(2,6,5,8)(19,26)(22,30)(27,32)(29,31)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,d^2=e,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, 125476);