#7193292

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=g,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=dhi,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, 7100488);