#288672

order	512
nilpotency class	3
isoclinism	isoclinic class of #288643
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2428
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3245	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=fh,b^2=efi,d^2=ei,e^2=h,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=dh,g^b=gi,f^c=fh,g^c=ghi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 195868);