#7224238

order	512
nilpotency class	3
isoclinism	isoclinic class of #7200022
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

1636 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)(19,24)(20,32)(21,27)(22,38)(23,28)(25,34)(26,35)(29,46)(30,41)(31,43)(33,40)(36,42)(37,45)(39,48)(44,47);(1,2)(3,6)(4,7)(5,8)(9,10)(11,16)(12,13)(14,15)(17,19)(18,24)(20,33)(21,36)(22,30)(23,31)(25,44)(26,46)(27,42)(28,43)(29,35)(32,40)(34,47)(37,48)(38,41)(39,45);(1,3,5,4)(2,6,8,7)(10,13)(14,16)(17,20)(18,25)(19,29)(21,34)(22,37)(23,35)(24,39)(26,27)(28,45)(30,47)(31,33)(32,38)(36,48)(40,42)(41,46)(43,44);(1,3,5,4)(2,7,8,6)(9,11,12,15)(10,14,13,16)(17,20)(18,26)(19,29)(21,37)(22,34)(23,35)(24,40)(25,27)(28,32)(30,48)(31,33)(36,47)(38,45)(39,42)(41,44)(43,46);(1,4,5,3)(2,7,8,6)(10,13)(14,16)(17,21,23,22)(18,27,28,38)(19,30,31,36)(20,34,35,37)(24,41,43,42)(25,26,45,32)(29,47,33,48)(39,46,44,40)

Polycyclic presentation:

2^9|a^2=f,c^2=f,d^2=fhi,e^2=fi,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, 7131434);