#261051

order	512
nilpotency class	3
isoclinism	isoclinic class of #248839
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	#2656	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1254 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)(5,6)(7,9)(8,10)(11,12)(13,14)(15,22)(16,20)(17,25)(18,21)(19,23)(24,27)(26,28);(1,2,4,3)(5,7)(6,9)(8,11)(10,12);(9,12)(13,15,18,24)(14,19,21,28)(16,23,17,26)(20,27,25,22);(1,3,4,2)(13,16,18,17)(14,20,21,25)(15,23,24,26)(19,27,28,22)

Polycyclic presentation:

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

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, 168247);