#7193282

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996639
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1663 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 52 points:

(2,4)(5,6,9,13)(7,11,14,10)(8,16,18,12)(15,19,17,20)(21,22,27,31)(23,28,26,40)(24,34,37,43)(25,30,39,46)(29,32,44,36)(33,41,38,45)(35,48,50,52)(42,47,51,49);(1,2)(3,4)(5,7,9,14)(6,10,13,11)(8,17,18,15)(12,20,16,19)(21,23,27,26)(22,28,31,40)(24,32,37,36)(25,33,39,38)(29,34,44,43)(30,41,46,45)(35,47,50,49)(42,48,51,52);(5,7)(6,11)(8,15)(9,14)(10,13)(12,20)(16,19)(17,18)(21,24,23,32,27,37,26,36)(22,29,40,43,31,44,28,34)(25,35,33,47,39,50,38,49)(30,42,45,52,46,51,41,48);(2,4)(7,14)(10,11)(15,17)(19,20)(21,25,27,39)(22,30,31,46)(23,33,26,38)(24,35,37,50)(28,41,40,45)(29,42,44,51)(32,47,36,49)(34,48,43,52);(5,8,9,18)(6,12,13,16)(7,15,14,17)(10,19,11,20)(21,23,27,26)(22,28,31,40)(24,36,37,32)(25,38,39,33)(29,43,44,34)(30,45,46,41)(35,47,50,49)(42,48,51,52)

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=gi,d^2=i,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7100478);