#6399113

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×D₄	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1706 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)(7,13)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20)(21,22,27,31)(23,28,34,43)(24,36,38,44)(25,30,26,39)(29,40,45,37)(32,46,48,51)(33,42,35,49)(41,50,52,47);(5,7)(6,10)(8,14)(9,15)(11,17)(12,13)(16,20)(18,19)(21,23,27,34)(22,28,31,43)(24,32,38,48)(25,33,26,35)(29,41,45,52)(30,42,39,49)(36,46,44,51)(37,47,40,50);(5,8,9,16)(6,11,12,18)(7,14,15,20)(10,17,13,19)(21,24,27,38)(22,29,31,45)(23,32,34,48)(25,37,26,40)(28,41,43,52)(30,44,39,36)(33,47,35,50)(42,51,49,46);(7,15)(10,13)(14,20)(17,19)(21,25,27,26)(22,30,31,39)(23,33,34,35)(24,37,38,40)(28,42,43,49)(29,44,45,36)(32,47,48,50)(41,51,52,46);(1,3,4,2)(6,12)(10,13)(11,18)(17,19)(23,34)(28,43)(32,48)(33,35)(41,52)(42,49)(46,51)(47,50)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=fi,d^2=i,e^2=hi,g^2=i,b^a=bf,c^a=cg,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, 6306309);