#6498933

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470901
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	#37305	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2104 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

Polycyclic presentation:

2^9|b^2=f,c^2=gh,e^2=h,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=dhi,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6406129);