#7200041

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	#35757	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2240 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)(3,7)(4,6)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18,23,27)(19,24,22,35)(20,30,32,37)(21,26,34,42)(25,36,40,28)(29,38,33,41)(31,44,46,47)(39,45,48,43);(1,3)(2,5)(4,8)(6,7)(9,11)(10,13)(12,15)(14,16)(17,19,23,22)(18,24,27,35)(20,28,32,36)(21,29,34,33)(25,37,40,30)(26,38,42,41)(31,43,46,45)(39,47,48,44);(11,15)(13,16)(17,20)(18,25)(19,28)(21,31)(22,36)(23,32)(24,37)(26,39)(27,40)(29,43)(30,35)(33,45)(34,46)(38,47)(41,44)(42,48);(3,8)(5,7)(10,14)(11,15)(17,21,23,34)(18,26,27,42)(19,29,22,33)(20,31,32,46)(24,38,35,41)(25,39,40,48)(28,43,36,45)(30,44,37,47);(2,6)(5,7)(11,15)(13,16)(17,19,23,22)(18,24,27,35)(20,28,32,36)(21,33,34,29)(25,37,40,30)(26,41,42,38)(31,45,46,43)(39,44,48,47)

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=i,e^2=i,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, 7107237);