#6376364

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

1690 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)(23,32)(24,36)(25,31)(26,30)(27,42)(28,35)(29,40)(33,48)(34,45)(37,51)(38,47)(39,50)(41,44)(43,52)(46,49);(1,3,4,2)(5,7)(6,11)(8,17)(9,15)(10,16)(12,20)(13,14)(18,19)(21,23)(22,28)(24,37)(25,41)(26,35)(27,34)(29,46)(30,32)(31,45)(33,39)(36,48)(38,49)(40,52)(42,44)(43,47)(50,51);(5,8,10,9)(6,12,13,18)(7,15,16,17)(11,19,14,20)(21,24,25,38,26,39,27,40)(22,29,42,50,30,47,31,36)(23,33,41,52,35,37,34,49)(28,43,44,48,32,46,45,51);(5,9,10,8)(6,12,13,18)(7,15,16,17)(11,20,14,19)(22,30)(28,32)(29,47)(31,42)(36,50)(43,46)(44,45)(48,51);(21,25,26,27)(22,31,30,42)(23,34,35,41)(24,38,39,40)(28,44,32,45)(29,36,47,50)(33,49,37,52)(43,48,46,51)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=fg,d^2=fi,e^2=i,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, 6283560);