#8927499

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

4004 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)(3,6)(4,13)(5,9)(7,10)(8,12)(11,16)(14,15)(17,18,24,27)(19,28,21,25)(20,23,22,26)(29,32,30,31)(33,34,39,36)(35,38,43,40)(37,41,46,44)(42,45,48,47);(1,3)(2,6)(4,5)(7,11)(8,14)(9,13)(10,16)(12,15)(17,19,24,21)(18,25,27,28)(20,29,22,30)(23,31,26,32)(33,34,39,36)(35,40,43,38)(37,44,46,41)(42,45,48,47);(3,11)(4,12)(6,16)(8,13)(17,20,24,22)(18,26,27,23)(19,29,21,30)(25,32,28,31)(33,35,39,43)(34,40,36,38)(37,42,46,48)(41,47,44,45);(1,4)(2,8)(3,5)(6,14)(7,12)(9,16)(10,13)(11,15)(17,21,24,19)(18,25,27,28)(20,29,22,30)(23,32,26,31)(33,36,39,34)(35,38,43,40)(37,44,46,41)(42,45,48,47);(1,5)(2,9)(3,12)(4,11)(6,8)(7,15)(10,14)(13,16)(17,22,24,20)(18,26,27,23)(19,30,21,29)(25,32,28,31)(33,37)(34,41)(35,42)(36,44)(38,45)(39,46)(40,47)(43,48);(1,6)(2,3)(4,14)(5,8)(7,16)(9,12)(10,11)(13,15)(17,23,24,26)(18,20,27,22)(19,31,21,32)(25,29,28,30)(33,38)(34,35)(36,43)(37,45)(39,40)(41,42)(44,48)(46,47)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=h,d^2=h,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, 8834695);