#402152

order	512
nilpotency class	3
isoclinism	isoclinic class of #396118
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×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	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

771 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)(3,4)(5,6,11,16,12,18,10,17)(7,15,20,29,21,28,9,13)(8,22,26,33,25,14,24,32)(19,30,35,27,34,31,23,36);(5,7,10,9,12,21,11,20)(6,13,16,15,18,29,17,28)(8,23,26,19,25,35,24,34)(14,30,33,36,22,31,32,27);(5,8,10,24,12,25,11,26)(6,14,16,32,18,22,17,33)(7,19,9,23,21,34,20,35)(13,27,15,31,29,36,28,30);(2,4)(5,9,10,21,12,20,11,7)(6,15,16,29,18,28,17,13)(8,23,24,34,25,35,26,19)(14,31,32,36,22,30,33,27)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 309348);