#6461369

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437753
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2310 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)(3,7)(4,5)(6,8)(9,10)(11,13)(12,14)(15,16)(17,18,22,26)(19,27,30,43)(20,32,34,24)(21,36,28,25)(23,37,39,31)(29,45,33,42)(35,41,44,47)(38,46,40,48);(2,5)(6,7)(9,11,12,15)(10,13,14,16)(17,19,28,37,22,30,21,31)(18,23,36,43,26,39,25,27)(20,33,44,46,34,29,35,48)(24,40,47,42,32,38,41,45);(1,3,4,8)(2,6,5,7)(10,14)(11,15)(17,20)(18,24)(19,29)(21,35)(22,34)(23,38)(25,41)(26,32)(27,42)(28,44)(30,33)(31,46)(36,47)(37,48)(39,40)(43,45);(11,15)(13,16)(19,30)(20,34)(23,39)(24,32)(27,43)(31,37)(35,44)(41,47);(19,30)(23,39)(27,43)(29,33)(31,37)(38,40)(42,45)(46,48)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6368565);