#218397

order	512
nilpotency class	3
isoclinism	isoclinic class of #30396
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂²×D₄	C₂²×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#97 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

541 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,d^2=eh,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cei,d^b=di,d^c=di,f^c=fh

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, 125593);