#495293

order	512
nilpotency class	3
isoclinism	isoclinic class of #362481
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2293
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#21270
prime radical	C₂×#135	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

506 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402489);