#6970649

order	512
nilpotency class	3
isoclinism	isoclinic class of #6907993
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×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1413 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)(3,7)(4,6)(5,8)(17,18,22,26)(19,27,29,43)(20,24,32,40)(21,34,35,25)(23,36,38,30)(28,42,31,46)(33,48,44,41)(37,47,39,45);(1,3,4,8)(2,5,6,7)(10,13)(11,15);(3,8)(5,7)(9,10)(11,14)(12,13)(15,16)(17,19,21,30,22,29,35,36)(18,23,25,27,26,38,34,43)(20,31,33,47,32,28,44,45)(24,39,41,46,40,37,48,42);(2,6)(5,7)(10,13)(14,16)(17,20)(18,24)(19,28)(21,33)(22,32)(23,37)(25,41)(26,40)(27,42)(29,31)(30,45)(34,48)(35,44)(36,47)(38,39)(43,46);(9,11,12,15)(10,14,13,16)(19,29)(20,32)(23,38)(24,40)(27,43)(30,36)(33,44)(41,48)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=g,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=dfhi,e^c=ei,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, 6877845);