#127695

order	512
nilpotency class	4
isoclinism	isoclinic class of #1357
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#331
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	C₂×#331
prime radical	C₂⁴	#100
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

504 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)(5,6)(7,16)(8,13)(9,22)(10,25)(11,15)(12,18)(14,23)(17,27)(19,33)(20,28)(21,29)(24,35)(26,34)(30,32)(31,36);(5,7,9,19,10,18,23,32,11,20,24,34,8,17,21,31)(6,12,14,26,13,16,22,30,15,27,29,33,25,28,35,36);(5,8,11,10)(6,13,15,25)(7,17,20,18)(9,21,24,23)(12,16,27,28)(14,22,29,35)(19,31,34,32)(26,30,33,36)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=i,d^2=g,f^2=gh,h^2=i,b^a=bd,d^a=dg,f^a=fgh,h^a=hi,d^b=dh,g^b=gi,h^b=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224797, #72224798, #72224799, …
maximal normal subgroup of	#72224797, #72224798, #72224799, …
maximal quotient of	C₄×#7268

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 34891);