#8253891

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	#41028	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4107 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)(3,7)(4,12)(5,6)(8,13)(9,11)(10,15)(14,16)(17,18,24,28)(19,29,31,25)(20,21,22,26)(23,27,32,30)(34,39)(35,42)(36,44)(45,47);(1,2)(3,9)(4,12)(5,6)(7,11)(8,13)(10,14)(15,16)(17,19,24,31)(18,25,28,29)(20,30,22,27)(21,32,26,23)(33,34,38,39)(35,37,42,40)(36,41,44,46)(43,45,48,47);(2,6)(7,9)(8,12)(14,15)(17,20,24,22)(18,26,28,21)(19,30,31,27)(23,29,32,25)(33,35,38,42)(34,40,39,37)(36,43,44,48)(41,47,46,45);(1,3,5,11)(2,7,6,9)(4,10,13,16)(8,14,12,15)(17,21)(18,20)(19,23)(22,28)(24,26)(25,27)(29,30)(31,32)(33,34)(35,40)(36,41)(37,42)(38,39)(43,47)(44,46)(45,48);(1,4)(2,8)(3,10)(5,13)(6,12)(7,14)(9,15)(11,16)(17,22,24,20)(18,21,28,26)(19,30,31,27)(23,29,32,25)(33,36)(34,41)(35,43)(37,45)(38,44)(39,46)(40,47)(42,48);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,13)(15,16)(17,23)(18,27)(19,21)(20,25)(22,29)(24,32)(26,31)(28,30)(33,37,38,40)(34,42,39,35)(36,45,44,47)(41,48,46,43)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8161087);