#8930306

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3765 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8837502);