#216573

order	512
nilpotency class	3
isoclinism	isoclinic class of #163298
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#35005	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1086 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:

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

Polycyclic presentation:

2^9|a^2=g,d^2=eh,f^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=ceh,d^b=dhi,f^c=fi,g^c=ghi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 123769);