#6812007

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667900
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#2540	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1675 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=h,d^2=f,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,g^c=gi,e^d=ei

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6719203);