#8472902

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	#45530	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4077 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=g,d^2=h,e^2=gh,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, 8380098);