#7230418

order	512
nilpotency class	3
isoclinism	isoclinic class of #7200022
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#25553	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1459 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|b^2=i,c^2=h,d^2=fh,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=eh,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7137614);