#6524832

order	512
nilpotency class	3
isoclinism	isoclinic class of #6499780
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3520	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3144 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)(8,12)(9,11)(13,17)(14,16)(15,18)(19,20)(21,22)(23,30)(24,29)(25,33)(26,28)(27,32)(31,35)(34,36);(5,7)(6,10)(8,14)(9,13)(11,17)(12,16)(15,19)(18,20)(21,23,26,31)(22,27,28,36)(24,32,25,34)(29,35,33,30);(2,4)(6,11)(7,13)(8,15)(16,20)(22,28)(23,31)(29,33)(32,34);(5,8,9,15)(6,12,11,18)(7,14,13,19)(10,16,17,20)(21,24,26,25)(22,29,28,33)(23,32,31,34)(27,35,36,30);(7,13)(10,17)(14,19)(16,20)(23,31)(27,36)(30,35)(32,34)

Polycyclic presentation:

2^9|a^2=gh,b^2=i,d^2=h,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=eh,f^b=fi,d^c=dhi

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, 6432028);