#7224261

order	512
nilpotency class	3
isoclinism	isoclinic class of #7200042
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#23853	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1661 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=f,d^2=fhi,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=efi,g^a=gi,c^b=ch,d^b=dfh,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, 7131457);