#7501015

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

3145 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)(5,6,9,12,11,16,10,15)(7,17,20,29,21,13,18,26)(8,22,25,30,24,27,23,14)(19,33,32,28,35,36,34,31);(6,12)(7,18)(8,23)(9,10)(14,30)(15,16)(20,21)(24,25)(26,29)(28,33)(31,36)(32,34);(2,4)(5,7)(6,13)(8,19)(9,20)(10,18)(11,21)(12,26)(14,28)(15,29)(16,17)(22,36)(23,34)(24,35)(25,32)(27,33)(30,31);(2,4)(5,8)(6,14)(7,19)(9,23)(10,25)(11,24)(12,27)(13,28)(15,22)(16,30)(17,31)(18,32)(20,34)(21,35)(26,33)(29,36);(5,9,11,10)(6,12,16,15)(7,20,21,18)(8,23,24,25)(13,26,17,29)(14,27,30,22)(19,34,35,32)(28,33,31,36)

Polycyclic presentation:

2^9|a^2=fi,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, 7408211);