#342562

order	512
nilpotency class	3
isoclinism	isoclinic class of #334823
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#459	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

715 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)(3,7)(4,6)(5,8)(9,10,14,16,13,11,12,15)(17,18)(19,24)(20,25)(21,23)(22,26)(27,32)(28,30)(29,31);(1,3)(2,5)(4,8)(6,7)(18,23)(19,27)(20,22)(24,31)(28,29)(30,32);(9,11,13,10)(12,15,14,16)(17,19)(18,24)(20,27)(21,28)(22,29)(23,30)(25,32)(26,31);(2,6)(3,8)(9,10,13,11)(12,16,14,15)(17,20,21,22)(18,25,23,26)(19,27,28,29)(24,32,30,31)

Polycyclic presentation:

2^9|a^2=fh,c^2=h,d^2=hi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=defi,c^b=cg,d^b=de,g^b=gi,f^c=fh,g^c=gi,f^d=fh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 249758);