#404373

order	512
nilpotency class	3
isoclinism	isoclinic class of #393893
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1318 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:

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

Polycyclic presentation:

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

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, 311569);