#6465652

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438997
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3154	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3156 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

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

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, 6372848);