#8430447

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	#36795	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4156 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8337643);