#9923387

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4219 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=i,d^2=ghi,e^2=gh,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	#72219279, #72219283, #72219285
maximal quotient of	#72219275

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9830583);