#233643

order	512
nilpotency class	3
isoclinism	isoclinic class of #233642
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#37536	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1704 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:

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

Polycyclic presentation:

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

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, 140839);