#7501016

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

3139 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)(5,6,12,18,11,7,10,15)(8,16,23,19,22,13,21,29)(9,24,25,17,27,30,26,14)(20,33,34,28,36,32,35,31);(5,7,11,6)(8,19,22,29)(9,24,27,30)(10,15,12,18)(13,23,16,21)(14,26,17,25)(20,28,36,31)(32,34,33,35);(2,4)(5,8)(6,13)(7,16)(9,20)(10,21)(11,22)(12,23)(14,28)(15,19)(17,31)(18,29)(24,32)(25,34)(26,35)(27,36)(30,33);(2,4)(5,9)(6,14)(7,17)(8,20)(10,25)(11,27)(12,26)(13,28)(15,24)(16,31)(18,30)(19,32)(21,34)(22,36)(23,35)(29,33);(5,10,11,12)(6,15,7,18)(8,21,22,23)(9,25,27,26)(13,19,16,29)(14,24,17,30)(20,34,36,35)(28,32,31,33)

Polycyclic presentation:

2^9|a^2=fi,b^2=i,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, 7408212);