#9815008

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	#40229	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4065 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)(3,7,11,15)(4,9,14,13)(5,12,16,10)(17,18,22,24)(19,26,30,23)(20,29,31,25)(21,28,32,27)(33,34)(35,37)(36,39)(38,44)(40,42)(41,46)(43,47)(45,48);(1,3)(2,7)(4,12)(5,13)(6,11)(8,15)(9,16)(10,14)(17,19)(18,23)(20,28)(21,29)(22,30)(24,26)(25,32)(27,31)(33,35)(34,40)(36,37)(38,43)(39,42)(41,47)(44,45)(46,48);(2,8)(3,11)(9,13)(10,12)(18,24)(20,31)(23,26)(27,28)(34,36)(35,42)(41,44)(43,48);(1,3)(2,7)(4,12)(5,13)(6,11)(8,15)(9,16)(10,14)(17,20)(18,25)(19,28)(21,26)(22,31)(23,32)(24,29)(27,30)(33,36,39,34)(35,37,42,40)(38,44,46,41)(43,45,48,47);(1,4)(2,9)(3,12)(5,15)(6,14)(7,16)(8,13)(10,11)(17,19)(18,26)(20,28)(21,25)(22,30)(23,24)(27,31)(29,32)(33,37)(34,35)(36,42)(38,45)(39,40)(41,43)(44,48)(46,47);(1,5,6,16)(2,10,8,12)(3,13,11,9)(4,7,14,15)(17,21)(18,27)(19,29)(20,26)(22,32)(23,31)(24,28)(25,30)(33,38)(34,41)(35,43)(36,44)(37,45)(39,46)(40,47)(42,48)

Polycyclic presentation:

2^9|a^2=gi,d^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=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, 9722204);