#6437964

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

4613 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:

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

Polycyclic presentation:

2^9|a^2=f,c^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	#72316331
maximal normal subgroup of	#72316331

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6345160);