#7729065

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	#43032	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4417 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,d^2=hi,e^2=ghi,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7636261);