#111164

order	512
nilpotency class	3
isoclinism	isoclinic class of #111151
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1437
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#121
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

591 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=efi,d^2=hi,e^2=h,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gi,d^b=dh,e^b=ei,f^b=fi,d^c=di

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 18360);