#6376041

order	512
nilpotency class	3
isoclinism	isoclinic class of #60771
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

1722 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)(8,12)(9,11)(10,14)(15,19)(16,18)(17,20)(21,22,26,30)(23,31,32,44)(24,28,35,42)(25,37,39,29)(27,38,40,34)(33,45,46,52)(36,49,48,43)(41,50,51,47);(1,3,4,2)(5,7)(6,10)(8,15)(9,14)(11,13)(12,17)(16,20)(18,19)(23,32)(27,40)(31,44)(33,46)(34,38)(41,51)(45,52)(47,50);(7,14)(10,13)(15,20)(17,19)(21,23,26,32)(22,27,30,40)(24,33,35,46)(25,38,39,34)(28,41,42,51)(29,44,37,31)(36,50,48,47)(43,52,49,45);(6,11)(10,13)(12,18)(17,19)(21,24)(22,28)(23,33)(25,36)(26,35)(27,41)(29,43)(30,42)(31,45)(32,46)(34,47)(37,49)(38,50)(39,48)(40,51)(44,52);(5,8,9,16)(6,12,11,18)(7,15,14,20)(10,17,13,19)(24,35)(28,42)(33,46)(36,48)(41,51)(43,49)(45,52)(47,50)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6283237);