#238528

order	512
nilpotency class	3
isoclinism	isoclinic class of #232291
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	C₂³×C₄	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1148 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)(5,6,10,12)(7,8,11,9)(13,14,16,20)(15,22,23,28)(17,26,18,21)(19,25,27,24);(1,2)(3,4)(5,7)(6,8)(9,12)(10,11)(13,14)(15,22)(16,20)(17,21)(18,26)(19,25)(23,28)(24,27);(1,3,4,2)(6,11)(7,12)(8,9)(13,15,16,23)(14,19,20,27)(17,25,18,24)(21,28,26,22);(5,8,10,9)(6,7,12,11)(14,20)(15,23)(21,26)(24,25)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 145724);