#248700

order	512
nilpotency class	3
isoclinism	isoclinic class of #248647
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1120 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)(3,6)(4,7)(5,8)(9,13)(10,12)(11,14)(15,16)(17,18,23,20)(19,27,21,30)(22,29,32,26)(24,31,25,28);(1,3)(2,6)(4,9)(5,10)(7,13)(8,12)(11,15)(14,16)(17,19,23,21)(18,24,20,25)(22,31,32,28)(26,30,29,27);(6,12)(13,16)(17,20,23,18)(19,25,21,24)(22,26,32,29)(27,31,30,28);(1,4,5,11)(2,7,8,14)(3,9,10,15)(6,13,12,16)(17,21,23,19)(18,25,20,24)(22,28,32,31)(26,27,29,30)

Polycyclic presentation:

2^9|a^2=h,b^2=h,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,e^c=eh,e^d=eh,f^d=fh

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, 155896);