#10368645

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	#41322	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3963 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=gi,d^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi

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, 10275841);