#347574

order	512
nilpotency class	3
isoclinism	isoclinic class of #346346
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1222 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)(9,12)(13,14,18,21)(15,22,23,28)(16,20,17,25)(19,26,27,24);(1,2,3,4)(15,23)(19,27)(22,28)(24,26);(5,6)(7,9)(8,10)(11,12)(13,15)(14,19)(16,24)(17,26)(18,23)(20,28)(21,27)(22,25);(5,7)(6,9)(8,11)(10,12)(13,16,18,17)(14,20,21,25)(15,24,23,26)(19,28,27,22)

Polycyclic presentation:

2^9|a^2=h,b^2=e,d^2=h,f^2=h,b^a=be,c^a=cf,d^a=dg,f^a=fh,c^b=ch,f^c=fh,g^c=gi,f^d=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 254770);