#9913403

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	#2679	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3883 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)(3,11)(4,10)(5,8)(6,14)(7,9)(12,13)(15,16)(17,18,23,21)(19,25,20,24)(22,26,30,29)(27,32,28,31)(35,41)(38,45)(42,47)(46,48);(1,3,7,5)(2,8,9,11)(4,12,14,15)(6,13,10,16)(17,19,23,20)(18,24,21,25)(22,27,30,28)(26,31,29,32)(33,34,37,40)(35,38,41,45)(36,39,44,43)(42,46,47,48);(2,9)(4,14)(8,11)(12,15)(17,19,23,20)(18,25,21,24)(22,27,30,28)(26,32,29,31)(33,34)(35,38)(36,39)(37,40)(41,45)(42,46)(43,44)(47,48);(1,4)(2,10)(3,12)(5,15)(6,9)(7,14)(8,16)(11,13)(17,20,23,19)(18,24,21,25)(22,28,30,27)(26,31,29,32)(33,35)(34,38)(36,42)(37,41)(39,46)(40,45)(43,48)(44,47);(1,5,7,3)(2,8,9,11)(4,15,14,12)(6,13,10,16)(17,21,23,18)(19,24,20,25)(22,26,30,29)(27,32,28,31)(33,34)(35,38)(36,43)(37,40)(39,44)(41,45)(42,48)(46,47);(1,6,7,10)(2,4,9,14)(3,13,5,16)(8,12,11,15)(17,22,23,30)(18,26,21,29)(19,27,20,28)(24,31,25,32)(33,36)(34,39)(35,42)(37,44)(38,46)(40,43)(41,47)(45,48)

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=gh,d^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

Relationships

maximal normal subgroup of	#72219283
maximal quotient of	#72219271

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9820599);