#7500900

order	512
nilpotency class	3
isoclinism	isoclinic class of #30484
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

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

3138 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)(5,6,11,17,12,18,10,16)(7,19,20,31,22,30,21,13)(8,14,25,33,26,34,24,32)(9,27,23,35,28,36,29,15);(5,7)(6,13)(8,23)(9,25)(10,20)(11,21)(12,22)(14,27)(15,33)(16,19)(17,30)(18,31)(24,28)(26,29)(32,35)(34,36);(2,4)(5,8)(6,14)(7,9)(10,24)(11,25)(12,26)(13,15)(16,32)(17,33)(18,34)(19,27)(20,23)(21,29)(22,28)(30,36)(31,35);(2,4)(5,9)(6,15)(7,8)(10,23)(11,29)(12,28)(13,14)(16,27)(17,36)(18,35)(19,32)(20,24)(21,25)(22,26)(30,33)(31,34);(5,10,12,11)(6,16,18,17)(7,20,22,21)(8,24,26,25)(9,23,28,29)(13,19,31,30)(14,32,34,33)(15,27,35,36)

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,f^b=fi,h^b=hi,h^c=hi,f^d=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7408096);