#361635

order	512
nilpotency class	3
isoclinism	isoclinic class of #184991
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#379
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3499	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3564 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,h^b=hi,f^c=fi,h^c=hi,h^d=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, 268831);