#114877

order	512
nilpotency class	3
isoclinism	isoclinic class of #114876
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

1107 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:

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

Polycyclic presentation:

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

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, 22073);