#8919225

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	#58160	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4049 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,d^2=g,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8826421);