#7955127

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	#45486	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4327 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=g,d^2=i,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=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, 7862323);