#405098

order	512
nilpotency class	3
isoclinism	isoclinic class of #393999
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	C₂³×#41	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2476 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)(8,10)(11,12)(13,14,17,19)(15,18,22,27)(16,23,24,20)(21,28,25,26);(5,7,8,11)(6,9,10,12)(15,21)(16,24)(18,26)(20,23)(22,25)(27,28);(2,4)(13,15)(14,18)(16,25)(17,22)(19,27)(20,28)(21,24)(23,26);(6,10)(9,12)(14,19)(18,27)(20,23)(26,28)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 312294);