#193522

order	512
nilpotency class	3
isoclinism	isoclinic class of #23860
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂²×C₂²⋊C₄
prime radical	#34964	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1277 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=h,d^2=h,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=dhi,e^b=eh,g^b=gh,d^c=di

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 100718);