#465304

order	512
nilpotency class	3
isoclinism	isoclinic class of #455110
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
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

833 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|b^2=egi,c^2=ef,d^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=di,d^c=de,g^c=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 372500);