#6437944

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437935
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 ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4612 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72312788, #72312814, #72313601, …
maximal normal subgroup of	#72312788, #72312814, #72313601, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6345140);