#232245

order	512
nilpotency class	3
isoclinism	isoclinic class of #232235
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	#9962	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1600 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)(7,10)(8,11)(9,12)(13,14)(15,22)(16,27)(17,20)(18,21)(19,25)(23,24)(26,28);(6,10)(7,8)(11,12)(13,15)(14,19)(16,26)(17,25)(18,24)(20,23)(21,28)(22,27);(2,4)(5,6)(7,10)(8,11)(9,12)(13,14)(15,23)(16,20)(17,27)(18,21)(19,25)(22,24)(26,28);(5,7,9,8)(6,10,12,11)(15,24)(19,28)(22,23)(25,26)

Polycyclic presentation:

2^9|d^2=i,e^2=h,g^2=h,b^a=be,c^a=cf,e^a=eh,g^a=gi,c^b=cg,d^b=dh,e^b=eh,g^b=gh,e^c=ei,g^c=gh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 139441);