#357314

order	512
nilpotency class	2
isoclinism	isoclinic class of #2668
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

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

958 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=ef,d^2=e,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,c^b=cef,d^b=de,d^c=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, 264510);