#7815823

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	#48100	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4440 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=h,d^2=h,e^2=gh,f^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dh,f^c=fi,e^d=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, 7723019);