#8097111

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3991 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=gh,d^2=g,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi

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, 8004307);