#6464999

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438997
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	#25419	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1508 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,c^2=f,d^2=hi,e^2=gi,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, 6372195);