#147751

order	512
nilpotency class	4
isoclinism	isoclinic class of #13343
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	B(4,𝔽₂)
commutator	C₂²×D₄	C₂²×C₄
Frattini, cosocle	C₂²×#41	C₂³
socle	C₂²	#1889
prime radical	C₂×#3025	C₂
LCS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

1728 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)(5,6,9,14,10,11,21,24)(7,15,17,28,18,29,31,36)(8,13,20,27,19,23,32,35)(12,25,26,30,22,33,34,16);(2,4)(6,11)(16,30)(17,31)(27,35);(5,7)(6,12)(8,16)(9,17)(10,18)(11,22)(13,15)(14,26)(19,30)(20,25)(21,31)(23,29)(24,34)(27,28)(32,33)(35,36)

Polycyclic presentation:

2^9|a^2=g,g^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fh,h^a=hi,c^b=cf,e^b=eh,g^b=gh,h^c=hi,e^d=ei,g^d=gi,f^e=fi,g^f=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250315, #72257211, #72257225, …
maximal normal subgroup of	#72250315, #72257211, #72257225, …
maximal quotient of	#72264939, #72308105

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54947);