#7224217

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	#23871	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1644 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)(17,18,22,28)(19,30,33,24)(20,35,37,26)(21,29,23,27)(25,38,43,36)(31,40,47,45)(32,41,34,42)(39,46,44,48);(1,2)(3,6)(4,7)(5,8)(9,10)(11,16)(12,13)(14,15)(17,19,22,33)(18,24,28,30)(20,31,37,47)(21,32,23,34)(25,39,43,44)(26,40,35,45)(27,41,29,42)(36,46,38,48);(1,3,5,4)(2,6,8,7)(10,13)(14,16)(17,20)(18,25)(19,31)(21,36)(22,37)(23,38)(24,39)(26,29)(27,35)(28,43)(30,44)(32,46)(33,47)(34,48)(40,42)(41,45);(1,3,5,4)(2,7,8,6)(9,11,12,15)(10,14,13,16)(17,20)(18,26)(19,31)(21,38)(22,37)(23,36)(24,40)(25,29)(27,43)(28,35)(30,45)(32,48)(33,47)(34,46)(39,42)(41,44);(1,4,5,3)(2,7,8,6)(10,13)(14,16)(17,21,22,23)(18,27,28,29)(19,32,33,34)(20,36,37,38)(24,41,30,42)(25,35,43,26)(31,46,47,48)(39,45,44,40)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=fi,d^2=fhi,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dhi,e^a=ef,g^a=gi,c^b=ch,d^b=dfhi,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, 7131413);