#246169

order	512
nilpotency class	3
isoclinism	isoclinic class of #246159
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	C₂³×C₄	C₂⁴
prime radical	#16500	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1554 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|b^2=h,d^2=h,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, 153365);