#8927996

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	#52050	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3843 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)(3,6)(4,13)(5,9)(7,10)(8,12)(11,16)(14,15)(17,18,22,20)(19,25,27,23)(21,24,29,28)(26,31,32,30)(33,34)(35,39)(36,42)(37,41)(38,40)(43,45)(44,48)(46,47);(1,3)(2,6)(4,5)(7,11)(8,14)(9,13)(10,16)(12,15)(17,18,22,20)(19,23,27,25)(21,28,29,24)(26,31,32,30)(34,35)(37,45)(40,42)(44,46);(3,11)(4,12)(6,16)(8,13)(17,19)(18,23)(20,25)(21,26)(22,27)(24,30)(28,31)(29,32)(33,35,39,34)(36,42,38,40)(37,43,45,41)(44,48,46,47);(1,4)(2,8)(3,5)(6,14)(7,12)(9,16)(10,13)(11,15)(17,20,22,18)(19,25,27,23)(21,28,29,24)(26,31,32,30)(33,36,39,38)(34,40,35,42)(37,44,45,46)(41,47,43,48);(1,5)(2,9)(3,12)(4,11)(6,8)(7,15)(10,14)(13,16)(17,21,22,29)(18,24,20,28)(19,26,27,32)(23,30,25,31)(33,37,39,45)(34,41,35,43)(36,44,38,46)(40,47,42,48);(1,6)(2,3)(4,14)(5,8)(7,16)(9,12)(10,11)(13,15)(17,19)(18,23)(20,25)(21,26)(22,27)(24,30)(28,31)(29,32)(33,38,39,36)(34,42,35,40)(37,46,45,44)(41,48,43,47)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=g,d^2=h,e^2=h,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, 8835192);