#211114

order	512
nilpotency class	3
isoclinism	isoclinic class of #25140
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₄×#121
prime radical	C₂×AGL(1,ℤ/8)	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

599 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eh,d^2=g,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,c^b=ci,d^b=dhi,d^c=di,f^c=fh

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 118310);