#8505954

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	C₂×#127	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3959 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)(3,7)(4,8)(5,9)(6,10)(11,15)(12,13)(14,16)(17,18,23,28)(19,25,20,24)(21,22,27,26)(29,30,31,32)(36,44)(40,45)(43,48)(46,47);(1,3,6,9)(2,7,10,5)(4,12,14,11)(8,13,16,15)(17,19,23,20)(18,24,28,25)(21,31,27,29)(22,30,26,32)(33,34)(35,39)(36,45)(37,41)(38,42)(40,44)(43,46)(47,48);(1,3)(2,5)(4,11)(6,9)(7,10)(8,13)(12,14)(15,16)(17,19,23,20)(18,25,28,24)(21,29,27,31)(22,30,26,32)(33,35)(34,37)(36,43)(38,41)(39,42)(40,46)(44,48)(45,47);(1,3)(2,7)(4,11)(5,10)(6,9)(8,15)(12,14)(13,16)(17,20,23,19)(18,25,28,24)(21,31,27,29)(22,30,26,32)(33,35)(34,39)(36,43)(37,42)(38,41)(40,47)(44,48)(45,46);(1,4)(2,8)(3,11)(5,13)(6,14)(7,15)(9,12)(10,16)(17,21,23,27)(18,26,28,22)(19,29,20,31)(24,30,25,32)(33,36,38,44)(34,40,42,45)(35,43,41,48)(37,46,39,47);(1,5,6,7)(2,9,10,3)(4,13,14,15)(8,12,16,11)(17,22,23,26)(18,27,28,21)(19,30,20,32)(24,29,25,31)(33,37,38,39)(34,41,42,35)(36,46,44,47)(40,48,45,43)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=g,d^2=g,e^2=ghi,f^2=gi,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, 8413150);