#6477472

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	#35076	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3284 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)(23,33)(24,36)(25,32)(26,31)(27,30)(28,35)(29,34);(2,4)(5,7)(6,10)(8,15)(9,14)(11,19)(12,18)(13,16)(17,20)(21,23,27,34)(22,28,30,36)(24,26,35,25)(29,32,33,31);(5,8,9,16)(6,11,12,20)(7,13,14,15)(10,17,18,19)(21,24,27,35)(22,29,30,33)(23,25,34,26)(28,31,36,32);(7,14)(10,18)(13,15)(17,19)(23,34)(24,35)(28,36)(29,33);(6,12)(10,18)(11,20)(17,19)(22,30)(28,36)(29,33)(31,32)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72315557, #72315558, #72315587
maximal normal subgroup of	#72315557, #72315558, #72315587
maximal quotient of	#72341971, #72342155, #72342161

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6384668);