#6376361

order	512
nilpotency class	3
isoclinism	isoclinic class of #60768
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1698 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)(7,14)(8,12)(9,18)(10,13)(11,16)(15,20)(17,19)(21,22)(23,31)(24,35)(25,29)(26,40)(27,33)(28,39)(30,37)(32,46)(34,42)(36,50)(38,49)(41,51)(43,48)(44,52)(45,47);(1,3,4,2)(5,7)(6,11)(8,17)(9,15)(10,16)(12,20)(13,14)(18,19)(21,23)(22,27)(24,36)(25,33)(26,34)(28,43)(29,31)(30,42)(32,38)(35,46)(37,47)(39,51)(40,45)(41,44)(48,52)(49,50);(5,8,10,9)(6,12,13,18)(7,15,16,17)(11,19,14,20)(21,24,37,52,25,38,26,39)(22,28,40,49,29,44,30,35)(23,32,47,51,33,36,34,48)(27,41,45,46,31,43,42,50);(5,9,10,8)(6,12,13,18)(7,15,16,17)(11,20,14,19)(22,29)(27,31)(28,44)(30,40)(35,49)(41,43)(42,45)(46,50);(23,33)(27,31)(32,36)(34,47)(41,43)(42,45)(46,50)(48,51)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=fg,d^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=cf,e^b=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, 6283557);