#8505499

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	#54403	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4049 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8412695);