#316158

order	512
nilpotency class	3
isoclinism	isoclinic class of #316149
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2504
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#124
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

442 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=egi,b^2=ei,c^2=ef,d^2=eh,e^2=h,f^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=gh,c^b=cg,f^b=fh,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, 223354);