#395253

order	512
nilpotency class	3
isoclinism	isoclinic class of #395252
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2428
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1735 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|b^2=ei,d^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,g^b=gi,f^c=fi,g^c=gi,h^c=hi,f^d=fi,h^d=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, 302449);