#6672583

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	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1723 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=f,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,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, 6579779);