#6481819

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470185
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	C₂×#3119	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1816 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:

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

Polycyclic presentation:

2^9|b^2=fi,d^2=h,e^2=i,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, 6389015);