#330728

order	512
nilpotency class	3
isoclinism	isoclinic class of #329743
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

517 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=i,c^2=h,d^2=ehi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dhi,c^b=cg,d^b=dh,g^b=gi,d^c=de,f^c=fh,g^c=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 237924);