#120045

order	512
nilpotency class	3
isoclinism	isoclinic class of #99835
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

854 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=dh,c^2=d,e^2=hi,b^a=bd,c^a=ce,e^a=eh,f^a=fi,c^b=cf,e^b=ei,e^c=ehi,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 27241);