#8430439

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	#43247	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4160 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)(4,9)(5,14)(6,8)(7,11)(12,16)(13,15)(20,28)(24,31)(26,32)(29,30)(33,34,38,42)(35,43,45,39)(36,40,46,48)(37,41,47,44);(1,3,7,6)(2,8,11,10)(4,12,5,13)(9,15,14,16)(17,18)(19,23)(20,24)(21,25)(22,27)(26,30)(28,31)(29,32)(33,35,38,45)(34,39,42,43)(36,44,46,41)(37,40,47,48);(1,4,7,5)(2,9,11,14)(3,12,6,13)(8,15,10,16)(17,19)(18,22)(20,26)(21,27)(23,25)(24,29)(28,32)(30,31)(33,34)(35,39)(36,40)(37,41)(38,42)(43,45)(44,47)(46,48);(1,5,7,4)(2,9,11,14)(3,13,6,12)(8,15,10,16)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,32)(29,31)(33,34)(35,43)(36,40)(37,44)(38,42)(39,45)(41,47)(46,48);(1,6,7,3)(2,10,11,8)(4,13,5,12)(9,16,14,15)(17,20)(18,24)(19,26)(21,28)(22,29)(23,30)(25,31)(27,32)(33,36,38,46)(34,40,42,48)(35,44,45,41)(37,39,47,43);(1,2)(3,8)(4,9)(5,14)(6,10)(7,11)(12,15)(13,16)(17,18)(19,22)(20,24)(21,25)(23,27)(26,29)(28,31)(30,32)(33,37)(34,41)(35,40)(36,39)(38,47)(42,44)(43,46)(45,48)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=gh,d^2=gh,e^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, 8337635);