#499035

order	512
nilpotency class	4
isoclinism	isoclinic class of #498807
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#466
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#3067	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1056 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|c^2=e,d^2=i,e^2=i,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fh,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,d^c=di,f^c=fhi,h^c=hi

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, 406231);