#8746013

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	#57748	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3947 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=gi,d^2=i,e^2=h,f^2=gh,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, 8653209);