#497829

order	512
nilpotency class	3
isoclinism	isoclinic class of #3290
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	D₄²
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	D₄×C₄⋊C₈
prime radical	C₂×#135	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

636 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,c^2=f,e^2=i,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,d^b=de,e^b=ei,e^d=ei

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225416, #72225418, #72225419, …
maximal normal subgroup of	#72225416, #72225418, #72225419, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405025);