#8097711

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

3906 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=ghi,d^2=g,e^2=gh,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, 8004907);