#410050

order	512
nilpotency class	3
isoclinism	isoclinic class of #394061
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂×#534	C₂²
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1606 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)(7,10)(8,9)(11,12)(13,14)(15,18)(16,23)(17,20)(19,24)(21,25)(22,26)(27,28);(1,2,4,3)(6,9)(10,12)(13,15)(14,18)(16,21)(17,22)(19,27)(20,26)(23,25)(24,28);(13,16)(14,19,20,23)(15,21)(17,24)(18,25,26,27)(22,28);(1,3,4,2)(5,7,8,11)(6,10,9,12)(16,24)(19,23)(21,28)(25,27)

Polycyclic presentation:

2^9|b^2=h,c^2=g,d^2=e,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,f^b=fi,d^c=di,f^c=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 317246);