#290827

order	512
nilpotency class	3
isoclinism	isoclinic class of #268139
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

799 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 198023);