#8932204

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	#56629	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3912 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=h,d^2=gh,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8839400);