#229815

order	512
nilpotency class	3
isoclinism	isoclinic class of #30396
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂²×D₈	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₄⋊C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

679 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=eh,f^2=h,g^2=i,b^a=be,c^a=cf,f^a=fh,d^b=dei,d^c=di,f^c=fh

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, 137011);