#417037

order	512
nilpotency class	3
isoclinism	isoclinic class of #396117
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#555
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2495
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

640 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:

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

Polycyclic presentation:

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

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, 324233);