#295311

order	512
nilpotency class	3
isoclinism	isoclinic class of #272667
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#2856	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

921 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:

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

Polycyclic presentation:

2^9|a^2=ehi,b^2=i,c^2=fi,d^2=e,g^2=i,b^a=be,c^a=cf,d^a=de,g^a=gh,c^b=cg,f^b=fh,g^b=gi,g^c=ghi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 202507);