#196324

order	512
nilpotency class	3
isoclinism	isoclinic class of #196320
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1395 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)(3,6,10,14)(4,11,12,7)(5,8)(9,13)(15,16)(17,18,21,24,22,25,20,29)(19,23,27,30,28,31,26,32);(1,3,5,10)(2,6)(4,9,12,15)(7,13)(8,14)(11,16)(17,19,21,27,22,28,20,26)(18,23,24,30,25,31,29,32);(1,4)(2,7)(3,9)(5,12)(6,13)(8,11)(10,15)(14,16);(17,20,22,21)(18,24,25,29)(19,26,28,27)(23,30,31,32)

Polycyclic presentation:

2^9|a^2=g,b^2=egh,d^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fh,e^b=eh,f^b=fh,g^b=gh,e^c=eh,g^c=gh

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, 103520);