#364196

order	512
nilpotency class	3
isoclinism	isoclinic class of #214672
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1400 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=eg,c^2=i,d^2=egi,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,f^b=fi,h^b=hi,f^c=fi,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, 271392);