#8436624

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	#39278	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4074 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=ghi,d^2=gh,e^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, 8343820);