#395163

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
prime radical	#23407	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1728 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)(7,15)(8,19)(9,21)(13,28)(14,24)(16,31)(17,18)(22,26)(23,29)(25,35)(30,32);(1,2)(3,4)(5,6,9,11,10,14,21,24)(7,12,17,22,18,27,15,26)(8,13,20,23,19,29,33,28)(16,25,31,34,32,35,30,36);(2,4)(5,7)(6,12)(8,16)(9,17)(10,18)(11,22)(13,25)(14,27)(15,21)(19,32)(20,31)(23,34)(24,26)(28,36)(29,35)(30,33);(5,8,9,20,10,19,21,33)(6,13,11,23,14,29,24,28)(7,16,17,31,18,32,15,30)(12,25,22,34,27,35,26,36)

Polycyclic presentation:

2^9|b^2=e,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, 302359);