#233431

order	512
nilpotency class	3
isoclinism	isoclinic class of #233431
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1916 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)(7,9)(8,11)(10,12)(13,14)(15,23)(16,22)(17,21)(18,20)(19,24)(25,27)(26,28);(2,4)(5,6)(7,9)(8,11)(10,12)(13,15)(14,19)(16,26)(17,24)(18,25)(20,28)(21,27)(22,23);(6,9)(7,12)(10,11)(15,24)(16,17)(19,23)(21,22)(25,26)(27,28);(14,20)(15,25)(21,22)(24,26)

Polycyclic presentation:

2^9|e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fi,g^a=ghi,c^b=cg,d^b=dh,e^b=eh,f^b=fi,g^b=ghi,e^c=eh,f^c=fi,g^c=ghi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224774, #72224787, #72224788
maximal normal subgroup of	#72224774, #72224787, #72224788

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 140627);