#7514183

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514172
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

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

3096 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)(5,6,12,17)(7,18,21,23)(8,16,11,14)(9,24,28,15)(10,19,13,29)(20,35,36,32)(22,31,27,33)(25,30,34,26);(1,2)(3,4)(5,7,12,21)(6,13,17,10)(8,19,11,29)(9,25,28,34)(14,18,16,23)(15,32,24,35)(20,27,36,22)(26,33,30,31);(2,4)(5,8,12,11)(6,14,17,16)(7,19,21,29)(9,22,28,27)(10,23,13,18)(15,31,24,33)(20,34,36,25)(26,35,30,32);(2,4)(5,9,12,28)(6,15,17,24)(7,20,21,36)(8,22,11,27)(10,26,13,30)(14,31,16,33)(18,32,23,35)(19,34,29,25);(5,10)(6,7)(8,23)(9,26)(11,18)(12,13)(14,19)(15,20)(16,29)(17,21)(22,35)(24,36)(25,33)(27,32)(28,30)(31,34)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cgi,d^b=dh,f^b=fi,h^b=hi,h^d=hi,f^e=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, 7421379);