#404041

order	512
nilpotency class	3
isoclinism	isoclinic class of #395174
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2404
prime radical	#23443	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1430 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)(7,11)(8,12)(9,13)(10,14)(15,21)(16,26)(17,24)(18,23)(19,28)(20,29)(22,30)(25,32)(27,33)(31,34)(35,36);(1,2,3,4)(5,7)(6,11)(8,16)(9,18)(10,17)(12,26)(13,23)(14,24)(15,25)(19,30)(20,31)(21,32)(22,28)(27,35)(29,34)(33,36);(5,8)(6,12)(7,16)(9,20)(10,19)(11,22)(13,29)(14,28)(15,27)(17,30)(18,31)(21,33)(23,35)(24,26)(25,34)(32,36);(6,13,14,15)(8,19)(11,23,24,25)(12,27,28,29)(16,30)(20,33)(22,34,26,35)(31,36)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72284607, #72285010, #72285492, …
maximal normal subgroup of	#72284607, #72285010, #72285492, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 311237);