#8925926

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	#2678	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3851 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=h,d^2=gh,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,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, 8833122);