#298140

order	512
nilpotency class	3
isoclinism	isoclinic class of #271771
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#459	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1059 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)(7,9)(8,11)(10,12)(13,14,17,20)(15,22,24,19)(16,25,27,28)(18,23,26,21);(1,2,4,3)(5,6,8,11)(7,10,12,9)(13,15,17,24)(14,19,20,22)(16,21,27,23)(18,25,26,28);(5,7,10,11,8,12,9,6)(13,16,26,22,17,27,18,19)(14,15,23,28,20,24,21,25);(1,3,4,2)(6,11)(7,12)(14,20)(15,24)(21,23)(25,28)

Polycyclic presentation:

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

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, 205336);