#370626

order	512
nilpotency class	2
isoclinism	isoclinic class of #16249
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

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

1106 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)(15,18)(16,20)(17,19)(21,22,27,23,26,30,25,29)(24,32,34,33,35,28,36,31);(1,2)(3,4)(6,10)(11,12)(13,15,17,16)(14,18,19,20)(21,23,27,30,26,29,25,22)(24,33,34,28,35,31,36,32);(2,4)(5,7,9,8)(6,11,10,12)(13,16,17,15)(14,18,19,20)(21,24,26,35)(22,28,30,32)(23,31,29,33)(25,36,27,34);(5,8,9,7)(6,11,10,12)(21,25,26,27)(22,29,30,23)(24,34,35,36)(28,31,32,33)

Polycyclic presentation:

2^9|a^2=h,b^2=fgh,c^2=efg,d^2=e,h^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,d^c=di

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 277822);