#392756

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=ei,d^2=i,f^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fi,d^b=di,f^c=fi

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, 299952);