#248806

order	512
nilpotency class	3
isoclinism	isoclinic class of #248743
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1253 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 156002);