#6469692

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437418
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 ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1591 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)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,18,22,26)(19,27,29,42)(20,31,33,24)(21,35,37,25)(23,36,39,30)(28,44,32,43)(34,41,48,46)(38,45,40,47);(9,10)(11,15)(12,13)(14,16)(17,19)(18,23)(20,32)(21,36)(22,29)(24,40)(25,42)(26,39)(27,35)(28,33)(30,37)(31,38)(34,45)(41,43)(44,46)(47,48);(2,6)(7,8)(10,13)(14,15)(17,20)(18,24)(19,28)(21,34)(22,33)(23,38)(25,41)(26,31)(27,43)(29,32)(30,45)(35,46)(36,47)(37,48)(39,40)(42,44);(1,3,5,4)(2,7,6,8)(9,11,12,16)(10,14,13,15)(19,29)(20,33)(23,39)(24,31)(27,42)(30,36)(34,48)(41,46);(1,4,5,3)(2,8,6,7)(20,33)(24,31)(28,32)(34,48)(38,40)(41,46)(43,44)(45,47)

Polycyclic presentation:

2^9|a^2=i,d^2=gi,e^2=gh,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,f^b=fi,d^c=di,e^c=ei

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6376888);