#6465639

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438949
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×#3150	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3134 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=gh,d^2=hi,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,f^b=fi,h^b=hi,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, 6372835);