#6399005

order	512
nilpotency class	3
isoclinism	isoclinic class of #61629
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 ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1709 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)(21,22,26,31)(23,27,33,40)(24,29,35,43)(25,28,37,30)(32,39,46,51)(34,38,47,41)(36,42,48,44)(45,49,52,50);(5,7)(6,10)(8,14)(9,15)(11,17)(12,13)(16,20)(18,19)(21,23,26,33)(22,27,31,40)(24,32,35,46)(25,34,37,47)(28,38,30,41)(29,39,43,51)(36,45,48,52)(42,49,44,50);(5,8,9,16)(6,11,12,18)(7,14,15,20)(10,17,13,19)(22,28)(25,37)(27,38)(29,42)(30,31)(34,47)(36,48)(39,49)(40,41)(43,44)(45,52)(50,51);(7,15)(10,13)(14,20)(17,19)(21,24)(22,29)(23,32)(25,36)(26,35)(27,39)(28,42)(30,44)(31,43)(33,46)(34,45)(37,48)(38,49)(40,51)(41,50)(47,52);(1,3,4,2)(6,12)(10,13)(11,18)(17,19)(23,33)(24,35)(27,40)(29,43)(34,47)(36,48)(38,41)(42,44)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=f,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,e^d=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6306201);