#8135693

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	#39048	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4243 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)(3,10)(4,8)(5,15)(6,16)(7,9)(11,12)(13,14)(20,28)(24,31)(26,32)(29,30)(33,34)(35,43)(36,41)(37,38)(39,42)(40,47)(44,46)(45,48);(3,4)(8,10)(11,13)(12,14)(17,18)(19,23)(20,24)(21,25)(22,27)(26,30)(28,31)(29,32)(33,35)(34,40)(36,44)(37,45)(38,46)(39,47)(41,48)(42,43);(1,3,7,4)(2,8,9,10)(5,11,16,13)(6,12,15,14)(17,19)(18,22)(20,26)(21,27)(23,25)(24,29)(28,32)(30,31)(33,36,39,37)(34,41,42,38)(35,44,47,45)(40,48,43,46);(1,4,7,3)(2,8,9,10)(5,11,16,13)(6,14,15,12)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,32)(29,31)(33,37,39,36)(34,41,42,38)(35,45,47,44)(40,48,43,46);(1,5)(2,6)(3,11)(4,13)(7,16)(8,12)(9,15)(10,14)(17,18)(19,22)(20,24)(21,25)(23,27)(26,29)(28,31)(30,32)(33,38,39,41)(34,37,42,36)(35,46,47,48)(40,45,43,44);(1,6)(2,5)(3,12)(4,14)(7,15)(8,11)(9,16)(10,13)(17,20)(18,24)(19,26)(21,28)(22,29)(23,30)(25,31)(27,32)(33,35)(34,40)(36,44)(37,45)(38,46)(39,47)(41,48)(42,43)

Polycyclic presentation:

2^9|c^2=gh,d^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^d=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, 8042889);