#466685

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

852 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)(5,6,8,9)(7,10,11,12)(13,14)(15,19)(16,26)(17,20)(18,21)(22,23)(24,28)(25,27);(6,9)(10,12)(13,15,23,28,17,25,18,26)(14,19,22,24,20,27,21,16);(1,2,4,3)(5,7)(6,10)(8,11)(9,12)(13,14,17,20)(15,24,25,16)(18,22,23,21)(19,26,27,28);(1,3,4,2)(7,11)(10,12)(13,16)(14,15)(17,24)(18,27)(19,23)(20,25)(21,26)(22,28)

Polycyclic presentation:

2^9|a^2=e,b^2=gi,c^2=f,d^2=fi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^c=de,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, 373881);