#6963047

order	512
nilpotency class	3
isoclinism	isoclinic class of #6884841
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×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1215 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)(3,11)(4,5)(6,9)(7,10)(8,14)(12,15)(13,16)(19,25)(22,26)(27,28)(30,31)(33,34,36,40,37,41,44,39)(35,38,43,45,42,46,48,47);(1,3)(2,8)(4,12)(5,13)(6,15)(7,14)(9,16)(10,11)(17,18)(19,26)(20,23)(21,24)(22,25)(27,31)(28,30)(29,32)(33,35,37,42)(34,38,41,46)(36,43,44,48)(39,47,40,45);(1,4,7,9)(2,5,10,6)(3,12,14,16)(8,13,11,15)(17,19,21,25)(18,22,24,26)(20,28,29,27)(23,31,32,30)(34,39)(35,42)(36,44)(38,45)(40,41)(46,47);(1,5,7,6)(2,9,10,4)(3,13,14,15)(8,16,11,12)(17,20,21,29)(18,23,24,32)(19,27,25,28)(22,30,26,31)(33,35)(34,38)(36,43)(37,42)(39,47)(40,45)(41,46)(44,48);(1,6,7,5)(2,9,10,4)(3,13,14,15)(8,12,11,16)(18,24)(22,26)(23,32)(30,31)(35,42)(38,46)(43,48)(45,47)

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,c^2=fhi,d^2=fhi,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,d^b=di,e^b=efh,d^c=dfh,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6870243);