#279225

order	512
nilpotency class	3
isoclinism	isoclinic class of #279211
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2416
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#41	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

521 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=e,c^2=hi,d^2=ei,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=di,g^b=gh,d^c=dh,f^c=fi,g^c=gh

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 186421);