#6498371

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470922
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	#37546	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1834 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:

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

Polycyclic presentation:

2^9|c^2=gh,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dh,e^b=ei,d^c=di,e^c=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, 6405567);