#265757

order	512
nilpotency class	3
isoclinism	isoclinic class of #246079
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

821 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)(3,7,4,6)(9,10)(11,15)(12,14)(13,16)(17,18,21,26)(19,20,29,24)(22,28,23,27)(25,31,32,30);(1,3,5,4)(2,6,8,7)(9,10)(11,15)(12,14)(13,16)(17,19,21,29)(18,24,26,20)(22,25,23,32)(27,31,28,30);(10,14)(11,12)(15,16)(17,20,21,24)(18,25,26,32)(19,28,29,27)(22,31,23,30);(1,4,5,3)(2,7,8,6)(9,11,13,12)(10,15,16,14)(18,26)(20,24)(27,28)(30,31)

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=hi,d^2=eh,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=gh,d^c=dh,f^c=fh,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, 172953);