#8750725

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

3743 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=h,d^2=gi,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8657921);