#7224796

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	#25793	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1655 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=gi,d^2=h,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^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, 7131992);