#6437966

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437963
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4745 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286845, #72316331
maximal normal subgroup of	#72286845, #72316331

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6345162);