#7500914

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500908
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

3150 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)(5,6,10,16,11,17,29,15)(7,18,20,30,21,36,19,12)(8,13,23,33,25,34,24,32)(9,26,27,35,28,31,22,14);(5,7,11,21)(6,12,17,30)(8,22,25,27)(9,23,28,24)(10,19,29,20)(13,31,34,26)(14,33,35,32)(15,18,16,36);(2,4)(5,8)(6,13)(7,9)(10,23)(11,25)(12,14)(15,32)(16,33)(17,34)(18,26)(19,22)(20,27)(21,28)(24,29)(30,35)(31,36);(2,4)(5,9)(6,14)(7,8)(10,22)(11,28)(12,13)(15,26)(16,31)(17,35)(18,32)(19,23)(20,24)(21,25)(27,29)(30,34)(33,36);(5,10,11,29)(6,15,17,16)(7,19,21,20)(8,23,25,24)(9,22,28,27)(12,18,30,36)(13,32,34,33)(14,26,35,31)

Polycyclic presentation:

2^9|a^2=f,b^2=i,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,c^b=ch,d^b=df,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, 7408110);