#299196

order	512
nilpotency class	3
isoclinism	isoclinic class of #299195
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2433
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#60643	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1221 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|b^2=e,c^2=f,d^2=eh,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=gh,c^b=cg,f^b=fh,g^c=gh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 206392);