#7049694

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996719
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

1321 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)(10,13)(11,15)(17,18,22,26)(19,27,28,42)(20,24,33,41)(21,34,35,25)(23,36,37,30)(29,43,44,48)(31,40,32,39)(38,45,47,46);(1,3,5,8)(2,6,4,7)(9,10,12,13)(11,16,15,14)(19,28)(23,37)(27,42)(29,44)(30,36)(38,47)(43,48)(45,46);(1,2,5,4)(3,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,19,21,30,22,28,35,36)(18,23,25,27,26,37,34,42)(20,29,32,46,33,44,31,45)(24,38,40,43,41,47,39,48);(3,8)(6,7)(10,13)(14,16)(17,20,22,33)(18,24,26,41)(19,29,28,44)(21,32,35,31)(23,38,37,47)(25,40,34,39)(27,43,42,48)(30,46,36,45);(1,4,5,2)(3,7,8,6)(9,11,12,15)(10,14,13,16)(17,21,22,35)(18,25,26,34)(19,30,28,36)(20,31,33,32)(23,27,37,42)(24,39,41,40)(29,45,44,46)(38,48,47,43)

Polycyclic presentation:

2^9|a^2=fh,b^2=f,c^2=fg,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=ehi,d^c=dhi,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6956890);