#148239

order	512
nilpotency class	4
isoclinism	isoclinic class of #13343
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

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	#1867
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

831 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)(5,6,10,14,11,15,9,24)(7,16,20,27,21,35,19,34)(8,13,22,31,23,32,25,30)(12,26,28,36,29,17,33,18);(5,7,8,18)(6,12,13,16)(9,19,25,17)(10,20,22,26)(11,21,23,36)(14,28,31,27)(15,29,32,35)(24,33,30,34);(2,4)(5,8)(6,13)(7,17)(9,25)(10,22)(11,23)(12,27)(14,31)(15,32)(16,33)(18,20)(19,36)(21,26)(24,30)(28,35)(29,34)

Polycyclic presentation:

2^9|a^2=g,b^2=d,f^2=i,g^2=i,b^a=bd,c^a=ce,f^a=fhi,c^b=cf,f^b=fhi,h^b=hi,d^c=dh,f^c=fi,f^d=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250465, #72257225, #72257229, …
maximal normal subgroup of	#72250465, #72257225, #72257229, …
maximal quotient of	#72265255, #72265268

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55435);