#184124

order	512
nilpotency class	2
isoclinism	isoclinic class of #2403
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂³×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂×C₄²	C₂⁴
commutator, socle	C₂³	C₂²×C₄²
prime radical	C₂²×D₄	C₄²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

827 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)(23,31)(24,28)(25,29)(26,30)(27,32)(33,34)(35,36);(2,4)(7,13)(10,16)(14,19)(17,20)(21,23,26,33,25,32,35,28)(22,27,30,24,29,31,36,34);(7,13)(10,16)(14,19)(17,20)(21,24,26,31,25,34,35,27)(22,28,30,23,29,33,36,32);(5,7,8,14,9,13,15,19)(6,10,11,17,12,16,18,20)(22,29)(23,32)(28,33)(30,36)

Polycyclic presentation:

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

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, 91320);