#183555

order	512
nilpotency class	2
isoclinism	isoclinic class of #2403
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂³×C₄

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

814 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 36 points:

(1,2,3,4)(5,6,8,11,9,12,15,18)(7,10,14,17,13,16,19,20)(21,22,25,31,27,30,26,32)(23,28,24,29,34,35,33,36);(2,4)(21,23,26,33,27,34,25,24)(22,28,32,36,30,35,31,29);(5,7,8,14,9,13,15,19)(6,10,11,17,12,16,18,20)(21,24,25,34,27,33,26,23)(22,29,31,35,30,36,32,28);(7,13)(10,16)(14,19)(17,20)(22,30)(28,35)(29,36)(31,32)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,f^2=h,g^2=i,b^a=be,d^a=di,d^c=dhi

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 90751);