#389272

order	512
nilpotency class	3
isoclinism	isoclinic class of #30484
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

872 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ei,c^2=efh,d^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,g^b=gi,f^c=fi,g^c=gi,h^c=hi,f^d=fi,h^d=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 296468);