#6815754

order	512
nilpotency class	3
isoclinism	isoclinic class of #6814827
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#61216	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1651 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=i,c^2=fgh,d^2=fi,e^2=h,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=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, 6722950);