#8091702

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	#43079	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4521 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)(9,10,15,17)(11,20,22,16)(12,18,21,14)(13,19,23,24)(25,26,30,29)(27,31,35,33)(28,36,37,32)(34,39,40,38);(1,2)(3,6)(4,5)(7,8)(9,11,15,22)(10,16,17,20)(12,13,21,23)(14,19,18,24)(25,26,30,29)(27,33,35,31)(28,32,37,36)(34,39,40,38);(10,17)(14,18)(16,20)(19,24)(26,29)(28,37)(31,33)(34,40);(9,12)(10,18)(11,13)(14,17)(15,21)(16,24)(19,20)(22,23)(25,27)(26,31)(28,34)(29,33)(30,35)(32,38)(36,39)(37,40);(9,13)(10,19)(11,21)(12,22)(14,20)(15,23)(16,18)(17,24)(25,28,30,37)(26,32,29,36)(27,34,35,40)(31,38,33,39);(1,3,4,7)(2,6,5,8)(9,14,15,18)(10,12,17,21)(11,19,22,24)(13,20,23,16)(25,29,30,26)(27,31,35,33)(28,36,37,32)(34,38,40,39)

Polycyclic presentation:

2^9|a^2=h,b^2=h,e^2=i,f^2=gi,b^a=bg,c^a=ch,e^a=ei,c^b=ci,d^b=di,e^c=ei,f^c=fh,e^d=ehi,f^d=fi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7998898);