#8155104

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	#2675	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3988 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=h,d^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8062300);