#494574

order	512
nilpotency class	4
isoclinism	isoclinic class of #2199
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	B(4,𝔽₂)
commutator	C₂×D₄	C₂³×C₄
Frattini, cosocle	C₂²×D₄	C₂⁴
socle	C₂²	C₂×B(4,𝔽₂)
prime radical	C₂×#3025	C₂
LCS	G ▷ C₂×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

2015 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,c^b=ci,f^b=fhi,e^c=eh,g^c=gh,f^e=fi,g^f=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72279139, #72279238, #72282724, …
maximal normal subgroup of	#72279139, #72279238, #72282724, …
maximal quotient of	#72287860, #72303866, #72304017

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401770);