#8930304

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

3971 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,d^2=ghi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8837500);