#395897

order	512
nilpotency class	3
isoclinism	isoclinic class of #393889
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1750 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=efi,b^2=e,c^2=ei,d^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=di,f^c=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 303093);