#450708

order	512
nilpotency class	3
isoclinism	isoclinic class of #417692
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	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

952 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

(2,6)(7,8)(9,10,13,15)(11,14,12,16)(17,18,23,27,22,26,31,32)(19,20,28,30,25,21,24,29);(1,2,5,6)(3,7,4,8)(9,11,13,12)(10,14,15,16)(17,19)(18,21)(20,26)(22,25)(23,28)(24,31)(27,29)(30,32);(1,3,5,4)(2,7,6,8)(9,12,13,11)(10,14,15,16)(17,20)(18,24)(19,27)(21,22)(23,30)(25,32)(26,28)(29,31);(1,4,5,3)(2,7,6,8)(17,21)(18,25)(19,26)(20,22)(23,30)(24,32)(27,28)(29,31)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 357904);