#207051

order	512
nilpotency class	3
isoclinism	isoclinic class of #162118
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

912 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=efg,c^2=fh,d^2=hi,e^2=hi,g^2=h,b^a=be,c^a=cf,e^a=eh,d^b=dh,e^b=ei,g^b=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 114247);