#355656

order	512
nilpotency class	2
isoclinism	isoclinic class of #2668
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂³×C₄	C₂⁴
commutator	C₂³	C₂²×C₄²
socle	C₂⁴	C₂³×C₄
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1302 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=eg,d^2=f,g^2=i,b^a=be,c^a=cf,d^a=de,c^b=cei,d^b=di

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 262852);