#370291

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	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

838 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=efgh,c^2=g,d^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gi,f^c=fi,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, 277487);