#9910991

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	#2464	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3812 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,d^2=g,e^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9818187);