#6481879

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470395
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#35259	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1854 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^9|b^2=f,d^2=h,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,e^c=ei,e^d=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, 6389075);