#400170

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	C₂²×#135	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1466 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)(13,14,17,23,19,25,18,24)(15,26,29,39,31,38,30,20)(16,32,33,21,35,41,34,22)(27,42,43,40,44,36,28,37);(2,4)(6,9)(10,12)(13,15,19,31)(14,20,25,39)(16,28,35,43)(17,30,18,29)(21,40,22,37)(23,38,24,26)(27,33,44,34)(32,36,41,42);(2,4)(13,16)(14,21)(15,27)(17,34)(18,33)(19,35)(20,36)(22,25)(23,32)(24,41)(26,40)(28,29)(30,43)(31,44)(37,38)(39,42);(5,7,8,11)(6,10,9,12)(13,16)(14,22)(15,28)(17,33)(18,34)(19,35)(20,37)(21,25)(23,32)(24,41)(26,42)(27,29)(30,44)(31,43)(36,38)(39,40)

Polycyclic presentation:

2^9|a^2=e,b^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,h^a=hi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,e^c=ei,f^c=fi,g^c=gi,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, 307366);