#93078

order	512
nilpotency class	4
isoclinism	isoclinic class of #93073
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1296
commutator	C₄×C₈	C₄²
Frattini, cosocle	C₂×C₈²	C₂²
prime radical	C₂³	#333
LCS	G ▷ C₄×C₈ ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₈² ▷ C₄² ▷ C₂² ▷ C₁

159 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 2 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=f,d^2=h,e^2=fghi,f^2=h,g^2=i,b^a=bc,c^a=cf,e^a=efg,f^a=fh,g^a=gi,c^b=cg,d^b=dh,f^b=fi,g^b=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 274);