#215275

order	512
nilpotency class	3
isoclinism	isoclinic class of #24378
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

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	C₂²×C₂²⋊C₄
prime radical	#2856	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1074 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)(9,10,14,15,13,11,12,16)(17,18)(19,27)(20,26)(21,23)(22,25)(24,29)(28,32)(30,31);(1,3,4,8)(2,5,6,7)(10,11)(15,16)(18,23)(19,28)(25,26)(29,30);(3,8)(5,7)(9,11)(10,13)(12,15)(14,16)(17,19)(18,24)(20,29)(21,28)(22,30)(23,31)(25,27)(26,32);(2,6)(5,7)(9,12,13,14)(10,15,11,16)(17,20,21,22)(18,25,23,26)(19,29,28,30)(24,27,31,32)

Polycyclic presentation:

2^9|a^2=g,b^2=eh,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, 122471);