#377688

order	512
nilpotency class	3
isoclinism	isoclinic class of #24808
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2416
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

915 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=e,d^2=fhi,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,g^b=gi,f^c=fi,g^c=gi,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 284884);