#2227914

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

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

2238 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=fghi,b^2=fg,c^2=fi,d^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72267714, #72267719
maximal normal subgroup of	#72267714, #72267719
maximal quotient of	#72267735, #72267740

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2135110);