#405029

order	512
nilpotency class	3
isoclinism	isoclinic class of #396611
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#124
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2398
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

685 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)(5,6,12,18,10,16,11,17)(7,8,14,22,20,13,25,26)(9,15,27,32,23,30,24,31)(19,21,29,34,33,28,35,36);(5,7)(6,13,16,8)(9,19)(10,20)(11,14)(12,25)(15,28,30,21)(17,26,18,22)(23,33)(24,29)(27,35)(31,36,32,34);(5,8,10,13)(6,14)(7,18)(9,21,23,28)(11,22,12,26)(15,29)(16,25)(17,20)(19,32)(24,34,27,36)(30,35)(31,33);(2,4)(5,9,11,24,10,23,12,27)(6,15,17,31,16,30,18,32)(7,19,14,29,20,33,25,35)(8,21,22,34,13,28,26,36)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 312225);