#496515

order	512
nilpotency class	4
isoclinism	isoclinic class of #480662
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂²×D₄	C₂⁴
prime radical	#2404	C₂²
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

967 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=ei,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=ce,d^c=di,g^c=gh,g^f=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72311246, #72311272, #72311281, …
maximal normal subgroup of	#72311246, #72311272, #72311281, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403711);