#360412

order	512
nilpotency class	3
isoclinism	isoclinic class of #25140
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

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

952 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)(13,14,19,24,17,22,18,23)(15,25,29,36,27,20,28,37)(16,30,34,21,32,40,33,39)(26,41,43,38,31,44,42,35);(6,9)(10,11)(13,15,18,28,17,27,19,29)(14,20,23,36,22,25,24,37)(16,31,33,43,32,26,34,42)(21,38,30,41,39,35,40,44);(13,16,17,32)(14,21,22,39)(15,26,27,31)(18,34,19,33)(20,35,25,38)(23,40,24,30)(28,43,29,42)(36,41,37,44);(5,7,8,12)(6,10,9,11)(14,22)(20,25)(21,39)(23,24)(30,40)(35,38)(36,37)(41,44)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 267608);