#8449304

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	#38938	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4131 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=ghi,d^2=gh,e^2=ghi,f^2=g,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, 8356500);