#8324623

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	#53504	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4069 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:

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

Polycyclic presentation:

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

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, 8231819);