#217599

order	512
nilpotency class	3
isoclinism	isoclinic class of #24378
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄

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	#1999	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

771 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)(4,6)(5,8)(10,13)(11,14)(15,16)(17,18,21,25,22,20,27,24)(19,26,28,31,29,23,30,32);(1,3)(2,5)(4,8)(6,7)(9,10,12,13)(11,15,16,14)(17,19,27,30,22,29,21,28)(18,23,24,31,20,26,25,32);(3,8)(5,7)(9,11)(10,14)(12,16)(13,15)(17,20)(18,22)(19,23)(21,24)(25,27)(26,29)(28,32)(30,31);(2,6)(5,7)(9,10,12,13)(11,14,16,15)(17,21,22,27)(18,24,20,25)(19,28,29,30)(23,31,26,32)

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 124795);