#354860

order	512
nilpotency class	2
isoclinism	isoclinic class of #2403
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂³×C₄	C₂⁴
commutator	C₂³	C₂²×C₄²
socle	C₂⁴	C₂³×C₄
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1654 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=e,d^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,c^b=ci,d^b=di

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, 262056);