#9926495

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	#53375	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4019 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=g,d^2=g,e^2=i,f^2=i,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
maximal quotient of	#72219271

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9833691);