#205792

order	512
nilpotency class	3
isoclinism	isoclinic class of #205785
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#390
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1183 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=f,d^2=hi,e^2=hi,f^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fhi,d^b=dh,e^b=ei,g^b=gi,d^c=di

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 112988);