#413471

order	512
nilpotency class	3
isoclinism	isoclinic class of #400051
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₈	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1034 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)(6,11)(8,22)(9,12)(13,17)(14,29)(15,16)(18,32)(19,20)(23,25)(26,35)(27,28)(33,34);(5,6,12,16,10,15,9,11)(7,17,19,13,21,27,20,28)(8,14,25,30,23,29,22,24)(18,31,33,26,34,36,32,35);(5,7,12,19,10,21,9,20)(6,13,16,27,15,28,11,17)(8,18,25,33,23,34,22,32)(14,26,30,36,29,35,24,31);(1,2,3,4)(5,8,12,25,10,23,9,22)(6,14,16,30,15,29,11,24)(7,18,19,33,21,34,20,32)(13,26,27,36,28,35,17,31)

Polycyclic presentation:

2^9|b^2=ef,c^2=ef,d^2=hi,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,g^b=gi,e^c=ei,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, 320667);