#6890427

order	512
nilpotency class	3
isoclinism	isoclinic class of #6884969
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	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1497 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=i,c^2=f,d^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efhi,d^c=dfh,e^c=ei,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, 6797623);