#8093160

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4670 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^b=eghi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#72239330, #72239345
Fitting subgroup of	#72239330
maximal subgroup of	#72239255, #72239330
maximal normal subgroup of	#72239255, #72239330

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8000356);