#360100

order	512
nilpotency class	3
isoclinism	isoclinic class of #196338
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1607 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 267296);