#405309

order	512
nilpotency class	3
isoclinism	isoclinic class of #394058
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1693 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|b^2=h,c^2=e,d^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,f^b=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, 312505);