#381648

order	512
nilpotency class	3
isoclinism	isoclinic class of #29902
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2622
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14304
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

813 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 288844);