#8097118

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

3945 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)(3,10)(4,9)(5,8)(6,11)(7,12)(13,15)(14,16)(17,18,23,26)(19,20,21,24)(22,25,30,32)(27,28,29,31)(33,34,38,41)(35,39,42,46)(36,37,40,45)(43,44,47,48);(1,3,7,5)(2,8,12,10)(4,13,14,6)(9,11,16,15)(17,19,23,21)(18,24,26,20)(22,27,30,29)(25,31,32,28)(35,42)(39,46)(43,47)(44,48);(1,4,7,14)(2,9,12,16)(3,13,5,6)(8,11,10,15)(17,18)(19,20)(21,24)(22,25)(23,26)(27,28)(29,31)(30,32)(33,35)(34,39)(36,43)(37,44)(38,42)(40,47)(41,46)(45,48);(1,5,7,3)(2,10,12,8)(4,6,14,13)(9,15,16,11)(17,20,23,24)(18,19,26,21)(22,28,30,31)(25,27,32,29);(1,6)(2,11)(3,4)(5,14)(7,13)(8,16)(9,10)(12,15)(17,21,23,19)(18,24,26,20)(22,27,30,29)(25,28,32,31)(33,36)(34,37)(35,43)(38,40)(39,44)(41,45)(42,47)(46,48);(1,2)(3,8)(4,9)(5,10)(6,15)(7,12)(11,13)(14,16)(17,22,23,30)(18,25,26,32)(19,27,21,29)(20,28,24,31)(33,37)(34,40)(35,44)(36,41)(38,45)(39,47)(42,48)(43,46)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=gh,d^2=g,e^2=h,f^2=h,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, 8004314);