#7649942

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	#38024	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4536 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=h,d^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^c=fhi,e^d=ehi

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, 7557138);