#7224260

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	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1631 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)(17,18,23,29)(19,24,32,42)(20,33,36,26)(21,28,22,27)(25,38,44,35)(30,45,34,40)(31,46,37,41)(39,47,43,48);(1,2)(3,6)(4,7)(5,8)(9,10)(11,16)(12,13)(14,15)(17,19)(18,24)(20,34)(21,37)(22,31)(23,32)(25,43)(26,45)(27,46)(28,41)(29,42)(30,36)(33,40)(35,47)(38,48)(39,44);(1,3,5,4)(2,6,8,7)(10,13)(14,16)(17,20)(18,25)(19,30)(21,35)(22,38)(23,36)(24,39)(26,28)(27,33)(29,44)(31,47)(32,34)(37,48)(40,41)(42,43)(45,46);(1,3,5,4)(2,7,8,6)(9,11,12,15)(10,14,13,16)(17,20)(18,26)(19,30)(21,38)(22,35)(23,36)(24,40)(25,28)(27,44)(29,33)(31,48)(32,34)(37,47)(39,41)(42,45)(43,46);(1,4,5,3)(2,7,8,6)(10,13)(14,16)(17,21,23,22)(18,27,29,28)(19,31,32,37)(20,35,36,38)(24,41,42,46)(25,33,44,26)(30,47,34,48)(39,40,43,45)

Polycyclic presentation:

2^9|a^2=fi,c^2=f,d^2=fhi,e^2=fi,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, 7131456);