#8933101

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	#46996	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4780 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#72285368, #72285397
maximal subgroup of	#72285346, #72285368, #72285502
maximal normal subgroup of	#72285346, #72285368, #72285502

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8840297);