#7046302

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996639
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1309 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)(10,11)(12,16)(17,18,21,23)(19,27,29,40)(20,25,31,36)(22,34,33,26)(24,37,35,30)(28,41,42,48)(32,45,44,39)(38,47,46,43);(1,3)(2,6)(4,7)(5,8)(9,10)(11,13)(12,15)(14,16)(18,23)(20,31)(24,35)(26,34)(27,40)(28,42)(32,44)(43,47);(9,11,13,10)(12,15,16,14)(17,19,22,30,21,29,33,37)(18,24,26,27,23,35,34,40)(20,28,32,43,31,42,44,47)(25,38,39,41,36,46,45,48);(3,8)(6,7)(10,11)(14,15)(17,20,21,31)(18,25,23,36)(19,28,29,42)(22,32,33,44)(24,38,35,46)(26,39,34,45)(27,41,40,48)(30,43,37,47);(1,4,5,2)(3,7,8,6)(9,12,13,16)(10,14,11,15)(19,29)(20,31)(24,35)(25,36)(27,40)(30,37)(32,44)(39,45)

Polycyclic presentation:

2^9|a^2=fh,c^2=g,d^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6953498);