#114450

order	512
nilpotency class	3
isoclinism	isoclinic class of #114446
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1438
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×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 — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fh,c^b=cf,d^b=di,e^b=eh,f^b=fi,g^b=ghi,f^c=fi,g^c=gh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72267846
maximal normal subgroup of	#72267846
maximal quotient of	#72267862

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 21646);