#8093430

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	#52451	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3954 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)(3,10)(4,5)(6,7)(8,11)(9,14)(12,16)(13,15)(20,28)(24,29)(26,32)(30,31)(34,35)(36,43)(40,42)(44,48);(2,7)(4,9)(8,10)(12,15)(17,18)(19,25)(20,24)(21,22)(23,27)(26,31)(28,29)(30,32)(33,34,38,35)(36,41,43,39)(37,40,45,42)(44,47,48,46);(3,11)(8,10)(12,15)(13,16)(18,22)(23,25)(24,29)(30,31)(33,35,38,34)(36,41,43,39)(37,42,45,40)(44,47,48,46);(1,3)(2,8)(4,12)(5,13)(6,11)(7,10)(9,15)(14,16)(17,18)(19,23)(20,24)(21,22)(25,27)(26,30)(28,29)(31,32)(33,36,38,43)(34,39,35,41)(37,44,45,48)(40,46,42,47);(1,4,6,9)(2,5,7,14)(3,12,11,15)(8,13,10,16)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,32)(29,31)(33,37,38,45)(34,40,35,42)(36,44,43,48)(39,46,41,47);(1,5)(2,9)(3,13)(4,7)(6,14)(8,15)(10,12)(11,16)(17,20)(18,24)(19,26)(21,28)(22,29)(23,30)(25,31)(27,32)

Polycyclic presentation:

2^9|b^2=h,c^2=h,d^2=h,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^b=eghi,d^c=dghi

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, 8000626);