#6482734

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470185
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	#2541	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1512 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)(9,10,13,14)(11,15,12,16)(17,18,22,24)(19,29,20,26)(21,28,23,27)(25,31,32,30);(2,5)(6,8)(9,10)(11,16)(12,15)(13,14)(18,24)(19,20)(27,28)(30,31);(10,14)(15,16)(17,19,22,20)(18,25,24,32)(21,30,23,31)(26,27,29,28);(9,11,13,12)(10,15,14,16)(17,20,22,19)(18,26,24,29)(21,30,23,31)(25,27,32,28);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,15,14,16)(17,21,22,23)(18,27,24,28)(19,30,20,31)(25,29,32,26)

Polycyclic presentation:

2^9|a^2=f,c^2=i,d^2=h,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6389930);