#393477

order	512
nilpotency class	3
isoclinism	isoclinic class of #32884
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1632 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)(5,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,18,22,26)(19,24,27,32)(20,23,21,25)(28,30,29,31);(2,4)(5,7,8,6)(18,23)(20,21)(24,30)(25,26)(28,29)(31,32);(10,13)(14,16)(17,19)(18,24)(20,28)(21,29)(22,27)(23,30)(25,31)(26,32);(9,11,12,15)(10,14,13,16)(18,25)(19,27)(20,21)(23,26)(24,30)(31,32)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,d^2=f,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,g^a=gi,e^b=ei,g^b=gi,d^c=di,e^d=ei,g^d=gi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 300673);