#184722

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

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

638 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)(3,6)(4,7)(5,8)(9,10,12,16,13,15,14,11)(17,18,22,24,21,25,20,29)(19,26,27,31,28,23,30,32);(1,3,6,2)(4,7,5,8)(9,10,14,11,13,15,12,16)(17,19)(18,23)(20,30)(21,28)(22,27)(24,32)(25,26)(29,31);(1,4,6,5)(2,7,3,8)(9,11,14,15,13,16,12,10)(17,18,20,29,21,25,22,24)(19,26,30,32,28,23,27,31);(1,5,6,4)(2,7,3,8)(9,12,13,14)(10,11,15,16)(17,20,21,22)(18,24,25,29)(19,27,28,30)(23,31,26,32)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,d^2=e,f^2=h,g^2=i,b^a=be,c^a=chi,d^a=de,c^b=ce,d^b=dh,d^c=dh

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, 91918);