#288704

order	512
nilpotency class	3
isoclinism	isoclinic class of #288683
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

573 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)(19,23)(20,25)(21,24)(22,26)(27,32)(28,30)(29,31);(2,8,11,10)(3,13,14,16)(4,12)(9,15)(17,19,21,29)(18,23,24,31)(20,28,22,27)(25,30,26,32);(1,3)(2,9)(4,10)(5,13)(6,16)(7,14)(8,12)(11,15)(18,24)(19,27)(20,22)(23,30)(28,29)(31,32);(1,4,5,15,7,12,6,9)(2,3,10,13,11,14,8,16)(17,20,21,22)(18,25,24,26)(19,28,29,27)(23,30,31,32)

Polycyclic presentation:

2^9|a^2=fhi,b^2=ef,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,d^c=di,f^c=fh,g^c=ghi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 195900);