#183807

order	512
nilpotency class	2
isoclinism	isoclinic class of #2668
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

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

714 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,d^2=e,f^2=h,g^2=i,b^a=be,d^a=dh,c^b=ch,d^b=dh,d^c=di

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, 91003);