#456729

order	512
nilpotency class	3
isoclinism	isoclinic class of #454426
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#537	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1024 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 363925);