#184420

order	512
nilpotency class	2
isoclinism	isoclinic class of #2668
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂²×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂×C₄²	C₂⁴
commutator, socle	C₂³	C₂²×C₄²
prime radical	C₂⁵	C₄²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1016 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)(9,10,13,11,12,14,16,15)(17,18,20,23,21,24,27,30)(19,22,26,29,25,28,31,32);(1,3)(2,5)(4,8)(6,7)(9,11,13,14,12,15,16,10)(17,19)(18,22)(20,26)(21,25)(23,29)(24,28)(27,31)(30,32);(3,8)(5,7)(10,14)(11,15);(2,6)(5,7)(10,14)(11,15)(19,25)(22,28)(26,31)(29,32)

Polycyclic presentation:

2^9|a^2=f,b^2=g,f^2=h,g^2=i,b^a=be,c^a=ci,d^a=de,c^b=ce,d^b=dh

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 91616);