#213658

order	512
nilpotency class	3
isoclinism	isoclinic class of #162550
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#36624	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1334 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,d^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=ce,f^c=fi,g^c=ghi

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, 120854);