#398060

order	512
nilpotency class	3
isoclinism	isoclinic class of #394267
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, prime radical, cosocle	C₂⁵	C₂⁴
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1610 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 305256);