#214076

order	512
nilpotency class	3
isoclinism	isoclinic class of #195972
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁶	C₂×C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁵×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2368 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 121272);