#7702857

order	512
nilpotency class	2
isoclinism	isoclinic class of #45438
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂⁵
commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	C₂⁴	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4226 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=g,d^2=gi,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dhi,e^c=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7610053);