#7046634

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

1299 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)(19,28)(20,31)(21,26)(22,27)(23,24)(25,32)(29,40)(30,41)(33,44)(34,37)(35,38)(36,42)(39,45)(43,48)(46,47);(1,3)(2,6)(4,7)(5,8)(9,10)(11,13)(12,15)(14,16)(17,19,22,23)(18,24,27,28)(20,32,34,29)(21,35,36,30)(25,31,40,37)(26,41,42,38)(33,43,46,45)(39,47,48,44);(9,11,13,10)(12,15,16,14)(17,20,19,29,22,34,23,32)(18,25,24,37,27,40,28,31)(21,33,30,43,36,46,35,45)(26,39,38,47,42,48,41,44);(3,8)(6,7)(10,11)(14,15)(17,21,22,36)(18,26,27,42)(19,30,23,35)(20,33,34,46)(24,38,28,41)(25,39,40,48)(29,43,32,45)(31,44,37,47);(1,4,5,2)(3,7,8,6)(9,12,13,16)(10,14,11,15)(18,27)(21,36)(24,28)(25,40)(30,35)(31,37)(33,46)(43,45)

Polycyclic presentation:

2^9|a^2=fh,b^2=i,c^2=gi,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, 6953830);