#184399

order	512
nilpotency class	2
isoclinism	isoclinic class of #2403
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

682 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)(9,10,13,11,12,15,14,16)(17,18,22,26,21,25,30,32)(19,27,28,20,24,23,31,29);(2,6)(7,8)(9,11,13,15,12,16,14,10)(17,19,21,24)(18,23,25,27)(20,26,29,32)(22,28,30,31);(1,3,5,4)(2,7,6,8)(9,11,14,10,12,16,13,15)(19,24)(20,29)(23,27)(28,31);(1,4,5,3)(2,8,6,7)(9,10,14,16,12,15,13,11)(17,20,30,27,21,29,22,23)(18,24,32,28,25,19,26,31)

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,d^a=di,c^b=ch,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, 91595);