#126959

order	512
nilpotency class	4
isoclinism	isoclinic class of #353
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#99
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂²×M₁₆	C₂³
socle	C₂²	#1459
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

636 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=h,d^2=i,e^2=h,h^2=i,b^a=bd,d^a=dg,f^a=fi,g^a=gi,e^b=egi,e^d=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239961, #72239962, #72239963, …
maximal normal subgroup of	#72239961, #72239962, #72239963, …
maximal quotient of	#72240001

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 34155);