#261384

order	512
nilpotency class	3
isoclinism	isoclinic class of #246079
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	#39359	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1240 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 168580);