#193472

order	512
nilpotency class	3
isoclinism	isoclinic class of #24384
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	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1386 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

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, 100668);