#395164

order	512
nilpotency class	3
isoclinism	isoclinic class of #395163
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1768 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:

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

Polycyclic presentation:

2^9|b^2=ei,d^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,f^c=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, 302360);