#6388232

order	512
nilpotency class	3
isoclinism	isoclinic class of #61609
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1754 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)(7,15)(8,12)(9,13)(10,14)(11,17)(16,20)(18,19)(21,22)(23,33)(24,37)(25,32)(26,30)(27,31)(28,35)(29,38)(34,48)(36,43)(39,46)(40,45)(41,44)(42,50)(47,51)(49,52);(1,3,4,2)(5,7)(6,11)(8,16)(9,18)(10,17)(12,19)(13,20)(14,15)(21,23,26,35)(22,28,30,33)(24,34,39,49)(25,40,27,36)(29,42,44,47)(31,45,32,43)(37,52,46,48)(38,51,41,50);(5,8,10,9)(6,12,14,13)(7,16,17,18)(11,19,15,20)(21,24,26,39)(22,29,30,44)(23,34,35,49)(25,38,27,41)(28,42,33,47)(31,37,32,46)(36,50,40,51)(43,48,45,52);(5,9,10,8)(6,13,14,12)(7,16,17,18)(11,19,15,20)(22,30)(23,35)(29,44)(31,32)(34,49)(36,40)(37,46)(50,51);(6,14)(11,15)(12,13)(19,20)(21,25,26,27)(22,31,30,32)(23,36,35,40)(24,38,39,41)(28,43,33,45)(29,37,44,46)(34,50,49,51)(42,48,47,52)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=f,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,g^a=gi,d^b=df,e^b=ei,g^c=gi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6295428);