#6438006

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437999
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	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×Q₈ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

4730 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)(7,15,18,11)(8,12)(9,21)(10,14)(13,23)(16,29,32,24)(17,25,34,30)(19,28)(20,27)(22,26)(31,35,33,36);(1,2)(3,4)(5,6)(7,11)(8,12)(9,13)(10,14)(15,18)(16,24)(17,25)(19,26)(20,27)(21,23)(22,28)(29,32)(30,34)(31,35)(33,36);(2,4)(5,7)(6,11)(8,16)(9,17)(10,18)(12,24)(13,25)(14,15)(19,31)(20,32)(21,30)(22,33)(23,34)(26,35)(27,29)(28,36);(2,4)(5,8,10,20)(6,12,14,27)(7,16,18,32)(9,22,23,19)(11,24,15,29)(13,28,21,26)(17,33,34,31)(25,36,30,35);(5,9,10,23)(6,13,14,21)(7,17,18,34)(8,19,20,22)(11,25,15,30)(12,26,27,28)(16,31,32,33)(24,35,29,36)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72317729, #72318728, #72318776
maximal normal subgroup of	#72317729, #72318728, #72318776

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6345202);