#6481528

order	512
nilpotency class	3
isoclinism	isoclinic class of #6469956
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3212 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)(23,27)(24,32)(25,28)(26,31)(29,34)(30,36)(33,35);(5,7)(6,10)(8,15)(9,13)(11,19)(12,17)(14,16)(18,20)(21,23,25,30)(22,26,28,35)(24,33,34,31)(27,29,36,32);(2,4)(7,13)(10,17)(14,15)(18,19)(21,23)(22,27)(24,31)(25,30)(26,32)(28,36)(29,35)(33,34);(5,8,9,16)(6,11,12,20)(7,14,13,15)(10,18,17,19)(22,28)(23,30)(29,32)(31,33);(6,12)(10,17)(11,20)(18,19)(22,28)(26,35)(27,36)(29,32)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72315541, #72315563, #72315567
maximal normal subgroup of	#72315541, #72315563, #72315567
maximal quotient of	#72316137, #72316237, #72316389

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6388724);