#10424692

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	#48319	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4080 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=hi,d^2=g,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,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, 10331888);