#254469

order	512
nilpotency class	3
isoclinism	isoclinic class of #254467
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#35438	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1043 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:

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

Polycyclic presentation:

2^9|b^2=e,c^2=i,g^2=h,b^a=be,c^a=cf,d^a=de,g^a=ghi,c^b=cg,f^b=fi,e^c=eh,g^c=gh,f^d=fh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 161665);