#252487

order	512
nilpotency class	3
isoclinism	isoclinic class of #246239
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

894 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 159683);