#147129

order	512
nilpotency class	3
isoclinism	isoclinic class of #11234
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#391
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1985
prime radical	C₂⁴	C₂×M₁₆
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

530 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,d^2=i,g^2=h,h^2=i,b^a=bd,c^a=ce,g^a=gi,c^b=ch,e^b=ei,f^b=fi,d^c=di,g^c=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239247, #72239249, #72239286, …
maximal normal subgroup of	#72239247, #72239249, #72239286, …
maximal quotient of	#72239342, #72279532

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54325);