#8508863

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	#2407	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3954 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=hi,d^2=gi,e^2=ghi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=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, 8416059);