#115762

order	512
nilpotency class	3
isoclinism	isoclinic class of #9914
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	#120
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

1204 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=hi,e^2=h,f^2=h,b^a=bd,c^a=ce,e^a=eh,f^a=fi,c^b=cf,e^b=ei,f^b=fh,e^c=eh,f^c=fh

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, 22958);