#6436541

order	512
nilpotency class	3
isoclinism	isoclinic class of #6432833
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	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3222 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)(13,16)(14,17)(15,18)(19,20)(21,22,26,29)(23,30,33,36)(24,28,25,34)(27,31,35,32);(5,7)(6,10)(8,14)(9,13)(11,16)(12,17)(15,19)(18,20)(21,23,26,33)(22,27,29,35)(24,32,25,31)(28,36,34,30);(2,4)(7,13)(10,16)(14,19)(17,20)(21,24,26,25)(22,28,29,34)(23,31,33,32)(27,30,35,36);(6,11)(10,16)(12,18)(17,20)(22,29)(27,35)(28,34)(30,36);(5,8,9,15)(6,12,11,18)(7,14,13,19)(10,17,16,20)(22,29)(27,35)(28,34)(30,36)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,e^2=hi,f^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ei,f^a=fi,c^b=ch,f^b=fi

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, 6343737);