#206034

order	512
nilpotency class	3
isoclinism	isoclinic class of #205751
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

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	C₂×#3169	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1075 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

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, 113230);