#362194

order	512
nilpotency class	3
isoclinism	isoclinic class of #24802
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

850 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 269390);