#400181

order	512
nilpotency class	3
isoclinism	isoclinic class of #393925
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1470 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 44 points:

(1,2,3,4)(5,6,8,9)(7,10,11,12)(13,14,19,24,20,16,18,23)(15,25,29,21,30,26,28,31)(17,33,36,22,35,38,34,32)(27,37,43,39,42,44,41,40);(2,4)(6,9)(10,12)(13,15,20,30)(14,21,16,31)(17,27,35,42)(18,29,19,28)(22,37,32,44)(23,26,24,25)(33,39,38,40)(34,43,36,41);(2,4)(13,16,20,14)(15,26,30,25)(17,32,35,22)(18,23,19,24)(21,28,31,29)(27,40,42,39)(33,36,38,34)(37,43,44,41);(5,7,8,11)(6,10,9,12)(13,17,20,35)(14,22,16,32)(15,27,30,42)(18,36,19,34)(21,37,31,44)(23,38,24,33)(25,39,26,40)(28,43,29,41)

Polycyclic presentation:

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

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, 307377);