#9717390

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	#48487	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4146 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)(3,7,10,5)(4,9,14,16)(11,13,12,15)(17,18)(19,29)(20,25)(21,26)(22,27)(23,28)(24,31)(30,32)(33,34,39,40)(35,43,38,41)(36,37,42,47)(44,45,46,48);(1,3,6,10)(2,7,8,5)(4,12,14,11)(9,15,16,13)(17,19,23,31)(18,24,28,29)(20,30,21,22)(25,27,26,32)(34,40)(37,47)(41,43)(45,48);(2,8)(3,10)(9,16)(11,12)(17,20,23,21)(18,25,28,26)(19,30,31,22)(24,27,29,32)(33,35,39,38)(34,41,40,43)(36,46,42,44)(37,45,47,48);(1,2)(3,7)(4,9)(5,10)(6,8)(11,13)(12,15)(14,16)(17,21,23,20)(18,26,28,25)(19,22,31,30)(24,32,29,27)(33,36,39,42)(34,37,40,47)(35,44,38,46)(41,48,43,45);(1,4,6,14)(2,9,8,16)(3,11,10,12)(5,15,7,13)(17,22)(18,27)(19,20)(21,31)(23,30)(24,26)(25,29)(28,32)(33,37)(34,42)(35,45)(36,40)(38,48)(39,47)(41,44)(43,46);(1,5,6,7)(2,10,8,3)(4,13,14,15)(9,11,16,12)(17,18)(19,24)(20,25)(21,26)(22,32)(23,28)(27,30)(29,31)(33,38,39,35)(34,43,40,41)(36,46,42,44)(37,45,47,48)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72219287
maximal normal subgroup of	#72219279, #72219283, #72219287
maximal quotient of	#72219273

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9624586);