#6498845

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470860
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#37618	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2154 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)(8,15)(9,11)(12,16)(13,17)(14,20)(18,19)(21,22)(23,24)(25,30)(26,36)(27,31)(28,32)(29,34)(33,35);(1,2,3,4)(5,7)(6,10)(8,14)(9,13)(11,17)(12,18)(15,20)(16,19)(21,23,26,32,27,33,25,29)(22,28,30,24,31,34,36,35);(6,11)(7,13)(12,15)(14,19)(21,24,25,28,27,35,26,34)(22,29,36,33,31,32,30,23);(7,13)(10,17)(14,19)(18,20)(21,25,27,26)(22,30,31,36)(23,32,33,29)(24,28,35,34);(5,8,9,16)(6,12,11,15)(7,14,13,19)(10,18,17,20)(22,31)(23,33)(29,32)(30,36)

Polycyclic presentation:

2^9|b^2=f,c^2=gh,d^2=i,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dh,e^b=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, 6406041);