#184393

order	512
nilpotency class	2
isoclinism	isoclinic class of #2403
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂²×C₈

Aliases

No aliases for this group.

Subgroups

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

676 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=eg,d^2=ef,f^2=h,g^2=i,b^a=be,c^a=ci,d^a=dh,c^b=ch,d^b=dh,d^c=dh

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 91589);