#7198595

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996355
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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1658 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=hi,c^2=gh,d^2=f,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7105791);