#8928754

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4075 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=gh,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi

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, 8835950);