#7501011

order	512
nilpotency class	3
isoclinism	isoclinic class of #7501009
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3209 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:

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

Polycyclic presentation:

2^9|a^2=f,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfg,c^b=ch,d^b=df,e^b=ei,f^b=fi,h^b=hi,h^c=hi,f^d=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7408207);