#8093169

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	C₂⁵	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4404 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)(11,19)(15,22)(20,21)(23,24)(25,26,30,34)(27,33,29,31)(28,32,38,36)(35,40,37,39);(2,5)(6,8)(9,10,14,18)(11,15,19,22)(12,17,13,16)(20,24,21,23)(25,26)(27,31)(28,36)(29,33)(30,34)(32,38)(35,40)(37,39);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,20)(13,21)(14,19)(16,23)(17,24)(18,22);(9,12,14,13)(10,16,18,17)(11,20,19,21)(15,23,22,24)(25,27,30,29)(26,31,34,33)(28,35,38,37)(32,39,36,40);(1,2)(3,6)(4,5)(7,8)(9,13,14,12)(10,17,18,16)(11,21,19,20)(15,24,22,23)(25,28)(26,32)(27,35)(29,37)(30,38)(31,39)(33,40)(34,36);(1,3)(2,6)(4,7)(5,8)(9,11)(10,15)(12,20)(13,21)(14,19)(16,23)(17,24)(18,22)(25,29,30,27)(26,33,34,31)(28,37,38,35)(32,40,36,39)

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=gh,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	#72238907, #72238908, #72239921
Fitting subgroup of	#72238907
maximal subgroup of	#72238888, #72238907, #72239901
maximal normal subgroup of	#72238888, #72238907, #72239901

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8000365);