#262223

order	512
nilpotency class	3
isoclinism	isoclinic class of #246847
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1061 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 169419);