#8455712

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	#2676	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3948 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)(3,12)(4,9)(5,15)(6,16)(7,11)(8,14)(10,13)(17,18,22,26)(19,27,28,23)(20,24,29,32)(21,25,31,30)(33,34,38,42)(35,43,46,39)(36,47,45,40)(37,41,44,48);(1,3,7,14)(2,8,11,12)(4,13,5,6)(9,16,15,10)(17,19)(18,23)(20,25)(21,24)(22,28)(26,27)(29,30)(31,32)(33,35,38,46)(34,39,42,43)(36,37,45,44)(40,41,47,48);(1,4,7,5)(2,9,11,15)(3,13,14,6)(8,16,12,10)(17,18)(19,23)(20,24)(21,25)(22,26)(27,28)(29,32)(30,31)(33,36,38,45)(34,40,42,47)(35,44,46,37)(39,48,43,41);(1,5,7,4)(2,9,11,15)(3,6,14,13)(8,16,12,10)(17,18)(19,27)(20,24)(21,30)(22,26)(23,28)(25,31)(29,32)(33,35)(34,39)(36,44)(37,45)(38,46)(40,48)(41,47)(42,43);(1,6)(2,10)(3,4)(5,14)(7,13)(8,9)(11,16)(12,15)(17,20,22,29)(18,24,26,32)(19,25,28,30)(21,27,31,23)(33,34)(35,39)(36,40)(37,41)(38,42)(43,46)(44,48)(45,47);(1,2)(3,8)(4,9)(5,15)(6,10)(7,11)(12,14)(13,16)(17,21,22,31)(18,25,26,30)(19,24,28,32)(20,27,29,23)(33,37,38,44)(34,41,42,48)(35,45,46,36)(39,47,43,40)

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=ghi,d^2=gh,e^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8362908);