#118016

order	512
nilpotency class	3
isoclinism	isoclinic class of #118014
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1439
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₂²⋊Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

557 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 25212);