#6481378

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470092
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	#35237	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3238 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)(8,11)(9,12)(13,17)(14,18)(15,19)(16,20)(21,22)(23,30)(24,28)(25,33)(26,29)(27,32)(31,36)(34,35);(2,4)(5,7)(6,10)(8,15)(9,13)(11,19)(12,17)(14,16)(18,20)(23,31)(27,35)(30,36)(32,34);(7,13)(10,17)(14,15)(18,19)(21,23,26,31)(22,27,29,35)(24,32,25,34)(28,36,33,30);(5,8,9,16)(6,11,12,20)(7,14,13,15)(10,18,17,19)(21,24,26,25)(22,28,29,33)(23,32,31,34)(27,36,35,30);(6,12)(10,17)(11,20)(18,19)(22,29)(27,35)(28,33)(30,36)

Polycyclic presentation:

2^9|c^2=i,d^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dhi,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72315557, #72315563, #72315587
maximal normal subgroup of	#72315557, #72315563, #72315587
maximal quotient of	#72341956, #72341963, #72342222

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6388574);