#7196698

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996375
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1802 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=i,c^2=ghi,d^2=f,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^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7103894);