#369182

order	512
nilpotency class	3
isoclinism	isoclinic class of #25446
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2428
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3155	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1862 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ehi,c^2=fg,d^2=ei,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,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, 276378);