#7779401

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	#2431	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4442 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=h,d^2=gh,e^2=h,f^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,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, 7686597);