#148236

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×#525	C₂²
LCS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

844 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)(5,6,9,14,10,13,21,27)(7,15,18,29,16,28,31,36)(8,12,20,25,19,26,32,35)(11,22,23,30,24,33,34,17);(5,7,8,17)(6,11,12,15)(9,18,20,22)(10,16,19,30)(13,24,26,28)(14,23,25,29)(21,31,32,33)(27,34,35,36);(2,4)(7,16)(15,28)(18,31)(29,36)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250688, #72257229, #72257245, …
maximal normal subgroup of	#72250688, #72257229, #72257245, …
maximal quotient of	#72265268, #72317728

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55432);