#215726

order	512
nilpotency class	3
isoclinism	isoclinic class of #184811
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	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1306 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)(3,7,8,5)(4,6)(9,10,13,14,12,16,11,15)(17,18)(19,25)(20,23)(21,24)(22,27)(26,31)(28,30)(29,32);(9,11,12,13)(10,14,16,15)(17,19,21,27)(18,22,24,25)(20,28,29,26)(23,31,32,30);(1,3)(2,5)(4,8)(6,7)(9,11,12,13)(10,15,16,14)(17,20,21,29)(18,23,24,32)(19,26,27,28)(22,30,25,31);(9,11,12,13)(10,14,16,15)(18,24)(22,25)(23,32)(30,31)

Polycyclic presentation:

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

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, 122922);