#112472

order	512
nilpotency class	3
isoclinism	isoclinic class of #112464
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1438
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

739 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 48 points:

(2,7)(3,10)(4,5)(9,14)(11,15)(12,13)(17,18,22,28,23,26,21,27)(19,29,31,44,33,24,32,40)(20,34,38,25,36,42,37,43)(30,46,48,39,47,41,35,45);(1,2,6,9)(3,11,13,15)(4,14,5,7)(8,10,16,12)(17,19)(18,24)(20,35)(21,31)(22,32)(23,33)(25,41)(26,29)(27,40)(28,44)(30,37)(34,45)(36,48)(38,47)(39,42)(43,46);(1,3,6,13)(2,8,9,16)(4,10,5,12)(7,15,14,11)(17,20,23,36)(18,25,26,43)(19,30,33,47)(21,38,22,37)(24,39,29,45)(27,42,28,34)(31,35,32,48)(40,46,44,41)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=h,d^2=i,e^2=h,f^2=h,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fh,c^b=cf,d^b=di,f^b=fh,g^b=ghi,d^c=dh,e^c=eh,f^c=fh,g^c=ghi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 19668);