#7708154

order	512
nilpotency class	2
isoclinism	isoclinic class of #47826
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	#52018	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4361 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)(3,7)(4,6)(5,8)(9,10)(11,18)(12,20)(13,15)(14,17)(16,21)(19,23)(22,24)(25,26,30,34)(27,31,35,38)(28,32,29,33)(36,39,37,40);(1,3,5,4)(2,6,8,7)(9,11,14,13)(10,15,17,18)(12,19,21,22)(16,23,20,24)(25,27)(26,31)(28,37)(29,36)(30,35)(32,40)(33,39)(34,38);(1,3,5,4)(2,7,8,6)(9,12,14,21)(10,16,17,20)(11,19,13,22)(15,23,18,24)(27,35)(28,29)(31,38)(32,33);(1,3,5,4)(2,7,8,6)(9,13,14,11)(10,15,17,18)(12,19,21,22)(16,24,20,23)(25,28,30,29)(26,32,34,33)(27,36,35,37)(31,39,38,40);(10,17)(15,18)(16,20)(23,24)(25,29,30,28)(26,33,34,32)(27,36,35,37)(31,39,38,40);(1,4,5,3)(2,6,8,7)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=g,d^2=gi,e^2=i,f^2=gh,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^b=di,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, 7615350);