#7205824

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	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1461 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:

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

Polycyclic presentation:

2^9|a^2=i,d^2=fi,e^2=f,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=ehi,e^c=ei,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, 7113020);