#248673

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	#61575	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1200 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)(9,14)(10,13)(11,15)(12,16)(18,23)(20,27)(21,29)(25,30)(26,31)(28,32);(1,3)(2,6)(4,9)(5,10)(7,14)(8,13)(11,16)(12,15)(17,18,22,25)(19,20,24,28)(21,30,29,23)(26,32,31,27);(1,4,5,11)(2,7,8,15)(3,9,10,16)(6,12,13,14)(17,19,22,24)(18,20,25,28)(21,31,29,26)(23,32,30,27);(17,20)(18,24)(19,25)(21,32)(22,28)(23,26)(27,29)(30,31)

Polycyclic presentation:

2^9|b^2=h,c^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, 155869);