#6376017

order	512
nilpotency class	3
isoclinism	isoclinic class of #60756
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	C₂×#121	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1762 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)(23,27)(24,34)(25,37)(26,29)(28,35)(30,39)(31,45)(32,41)(33,49)(36,44)(38,43)(40,47)(42,48)(46,52)(50,51);(1,3,4,2)(5,7)(6,10)(8,15)(9,14)(11,13)(12,17)(16,20)(18,19)(21,23)(22,27)(24,31)(25,33)(26,32)(28,40)(29,41)(30,42)(34,45)(35,47)(36,46)(37,49)(38,50)(39,48)(43,51)(44,52);(7,14)(10,13)(15,20)(17,19)(21,24,26,36)(22,28,29,43)(23,31,32,46)(25,38,39,35)(27,40,41,51)(30,44,37,34)(33,50,48,47)(42,52,49,45);(6,11)(10,13)(12,18)(17,19)(23,32)(27,41)(31,46)(33,48)(40,51)(42,49)(45,52)(47,50);(5,8,9,16)(6,12,11,18)(7,15,14,20)(10,17,13,19)(22,29)(23,32)(28,43)(30,37)(31,46)(33,48)(34,44)(47,50)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=cf,d^b=di,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, 6283213);