#8095031

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3843 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8002227);