#7234148

order	512
nilpotency class	3
isoclinism	isoclinic class of #7207866
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	#53078	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1466 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)(4,5)(7,8)(9,10)(11,16)(12,15)(13,14)(17,18,22,27)(19,23,30,26)(20,31,34,38)(21,35,37,25)(24,32,41,28)(29,42,36,39)(33,46,47,44)(40,43,48,45);(2,5)(3,7)(10,14)(11,12)(17,19,22,30)(18,23,27,26)(20,32,34,28)(21,36,37,29)(24,31,41,38)(25,42,35,39)(33,43,47,45)(40,46,48,44);(2,5)(6,8)(9,11,13,12)(10,15,14,16)(17,20,30,32,22,34,19,28)(18,24,23,38,27,41,26,31)(21,33,36,45,37,47,29,43)(25,40,39,46,35,48,42,44);(1,3,4,7)(2,6,5,8)(10,14)(15,16)(17,21,22,37)(18,25,27,35)(19,29,30,36)(20,33,34,47)(23,39,26,42)(24,40,41,48)(28,43,32,45)(31,44,38,46);(2,5)(6,8)(9,12,13,11)(10,16,14,15)(17,19,22,30)(18,23,27,26)(20,28,34,32)(21,36,37,29)(24,38,41,31)(25,42,35,39)(33,45,47,43)(40,44,48,46)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fgh,d^2=fh,e^2=h,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^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7141344);