#8497823

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	#53174	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4045 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)(3,11)(4,13)(5,10)(6,7)(8,12)(9,14)(15,16)(17,18,24,22)(19,28,23,25)(20,27,21,26)(29,31,30,32)(36,44)(40,45)(43,48)(46,47);(2,7)(4,14)(8,11)(10,16)(17,19,24,23)(18,25,22,28)(20,29,21,30)(26,31,27,32)(33,34)(35,39)(36,45)(37,41)(38,42)(40,44)(43,46)(47,48);(17,20,24,21)(18,26,22,27)(19,29,23,30)(25,31,28,32)(33,35)(34,37)(36,43)(38,41)(39,42)(40,46)(44,48)(45,47);(1,3,6,12)(2,8,7,11)(4,10,14,16)(5,13,15,9)(17,21,24,20)(18,27,22,26)(19,29,23,30)(25,31,28,32)(33,35)(34,39)(36,43)(37,42)(38,41)(40,47)(44,48)(45,46);(1,4)(2,9)(3,10)(5,8)(6,14)(7,13)(11,15)(12,16)(17,22,24,18)(19,25,23,28)(20,27,21,26)(29,31,30,32)(33,36)(34,40)(35,43)(37,46)(38,44)(39,47)(41,48)(42,45);(1,5)(2,10)(3,9)(4,8)(6,15)(7,16)(11,14)(12,13)(17,23,24,19)(18,25,22,28)(20,30,21,29)(26,31,27,32)(33,37,38,39)(34,41,42,35)(36,46,44,47)(40,48,45,43)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=h,d^2=g,e^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8405019);