#8127531

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	#57199	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4182 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=hi,d^2=h,e^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^c=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, 8034727);