#295116

order	512
nilpotency class	3
isoclinism	isoclinic class of #268139
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	C₂×C₄²	C₂⁴
prime radical	#10078	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

865 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=f,d^2=ei,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=ghi,g^c=gh

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 202312);