#6672449

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 ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1733 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)(21,22,27,33)(23,34,38,29)(24,31,25,40)(26,32,28,43)(30,41,48,39)(35,50,36,45)(37,47,42,46)(44,52,49,51);(1,2)(3,4)(5,7,9,15)(6,10,12,13)(8,14,17,20)(11,18,16,19)(21,23)(22,29)(24,35)(25,36)(26,42)(27,38)(28,37)(30,44)(31,45)(32,47)(33,34)(39,52)(40,50)(41,51)(43,46)(48,49);(2,4)(21,24,27,25)(22,30,33,48)(23,35,38,36)(26,39,28,41)(29,44,34,49)(31,32,40,43)(37,51,42,52)(45,47,50,46);(5,8,9,17)(6,11,12,16)(7,14,15,20)(10,18,13,19)(21,25,27,24)(22,31,33,40)(23,36,38,35)(26,39,28,41)(29,45,34,50)(30,32,48,43)(37,51,42,52)(44,47,49,46);(2,4)(7,15)(10,13)(14,20)(18,19)(21,26,27,28)(22,32,33,43)(23,37,38,42)(24,39,25,41)(29,46,34,47)(30,40,48,31)(35,51,36,52)(44,45,49,50)

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=i,d^2=f,e^2=i,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, 6579645);