#8455728

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3889 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=ghi,d^2=gh,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8362924);