#8127524

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	#50087	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4198 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)(17,18,22,24)(19,25,26,20)(21,29,30,23)(27,28,32,31)(33,34)(35,44)(36,37)(38,41)(39,40)(42,43)(45,48)(46,47);(1,3,7,5)(2,8,4,11)(6,10,14,13)(9,15,12,16)(17,18)(19,20)(21,23)(22,24)(25,26)(27,28)(29,30)(31,32)(34,40)(36,43)(41,44)(45,47);(2,4)(3,5)(9,12)(10,13)(19,26)(20,25)(27,32)(28,31)(33,35,39,38)(34,41,40,44)(36,47,43,45)(37,48,42,46);(1,4,7,2)(3,8,5,11)(6,12,14,9)(10,15,13,16)(17,19)(18,20)(21,27)(22,26)(23,28)(24,25)(29,31)(30,32)(33,36,39,43)(34,42,40,37)(35,45,38,47)(41,48,44,46);(1,5,7,3)(2,8,4,11)(6,13,14,10)(9,15,12,16)(17,20)(18,19)(21,28)(22,25)(23,27)(24,26)(29,32)(30,31)(33,37,39,42)(34,43,40,36)(35,46,38,48)(41,47,44,45);(1,6)(2,9)(3,10)(4,12)(5,13)(7,14)(8,15)(11,16)(17,21,22,30)(18,23,24,29)(19,27,26,32)(20,28,25,31)(33,38,39,35)(34,44,40,41)(36,47,43,45)(37,48,42,46)

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=hi,d^2=h,e^2=h,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, 8034720);