#7013200

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1377 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=i,c^2=f,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eh,d^c=dh,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, 6920396);