#296727

order	512
nilpotency class	3
isoclinism	isoclinic class of #271515
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#2537	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

831 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

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, 203923);