#8135331

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

3838 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=h,d^2=i,e^2=h,f^2=ghi,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, 8042527);