#405134

order	512
nilpotency class	3
isoclinism	isoclinic class of #393958
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3005	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2537 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:

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

Polycyclic presentation:

2^9|b^2=h,d^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,f^b=fi,f^c=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 312330);