#248802

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
prime radical	#18168	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1161 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)(26,30)(27,31);(1,3)(2,6)(4,9)(5,10)(7,13)(8,12)(11,15)(14,16)(17,18,22,26)(19,20,25,24)(21,30,29,23)(27,32,31,28);(4,11)(6,12)(7,14)(9,15)(17,19,22,25)(18,24,26,20)(21,31,29,27)(23,28,30,32);(1,4,5,11)(2,7,8,14)(3,9,10,15)(6,13,12,16)(17,20)(18,25)(19,26)(21,32)(22,24)(23,27)(28,29)(30,31)

Polycyclic presentation:

2^9|b^2=h,c^2=h,d^2=hi,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, 155998);