#246199

order	512
nilpotency class	3
isoclinism	isoclinic class of #246159
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

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

1716 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:

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

Polycyclic presentation:

2^9|c^2=i,e^2=h,b^a=be,c^a=cf,d^a=dei,e^a=eh,c^b=cg,e^b=eh,f^b=fi,e^c=ei,e^d=eh,f^d=fi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 153395);