#8750752

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

3786 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)(17,18,22,23)(19,20,25,28)(21,24,29,30)(26,27,31,32)(33,34,39,41)(35,40,36,44)(37,38,47,48)(42,46,45,43);(1,3,7,5)(2,8,11,10)(4,14,15,12)(6,13,9,16)(18,23)(19,25)(24,30)(26,31)(33,35,39,36)(34,40,41,44)(37,45,47,42)(38,43,48,46);(1,3,7,5)(2,8,11,10)(4,14,15,12)(6,13,9,16)(17,19)(18,20)(21,26)(22,25)(23,28)(24,27)(29,31)(30,32)(33,36,39,35)(34,40,41,44)(37,45,47,42)(38,46,48,43);(1,4,7,15)(2,9,11,6)(3,12,5,14)(8,13,10,16)(17,20)(18,19)(21,27)(22,28)(23,25)(24,26)(29,32)(30,31)(33,37)(34,38)(35,42)(36,45)(39,47)(40,46)(41,48)(43,44);(1,5,7,3)(2,10,11,8)(4,14,15,12)(6,13,9,16)(17,21,22,29)(18,24,23,30)(19,26,25,31)(20,27,28,32)(33,38)(34,37)(35,43)(36,46)(39,48)(40,45)(41,47)(42,44);(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=h,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, 8657948);