#7206371

order	512
nilpotency class	3
isoclinism	isoclinic class of #7200042
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	#23975	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1564 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)(3,6)(4,7)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18,23,27)(19,28,31,39)(20,32,33,25)(21,26,22,36)(24,37,40,30)(29,44,45,43)(34,42,35,41)(38,47,48,46);(1,3,6,2)(4,7,5,8)(9,11)(10,13)(12,15)(14,16)(20,33)(25,32)(29,45)(34,35)(38,48)(41,42)(43,44)(46,47);(11,15)(13,16)(17,19,23,31)(18,24,27,40)(20,29,33,45)(21,30,22,37)(25,38,32,48)(26,39,36,28)(34,46,35,47)(41,43,42,44);(1,4,6,5)(2,7,3,8)(10,14)(11,15)(17,20)(18,25)(19,29)(21,35)(22,34)(23,33)(24,38)(26,42)(27,32)(28,43)(30,47)(31,45)(36,41)(37,46)(39,44)(40,48);(1,5,6,4)(2,7,3,8)(11,15)(13,16)(17,21,23,22)(18,26,27,36)(19,30,31,37)(20,34,33,35)(24,39,40,28)(25,41,32,42)(29,46,45,47)(38,43,48,44)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=i,d^2=f,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=dhi,e^a=ef,g^a=gi,c^b=ch,d^b=dfhi,e^b=eh,g^c=gi,e^d=ei

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7113567);