#493434

order	512
nilpotency class	4
isoclinism	isoclinic class of #480274
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×D₄	C₂⁴
prime radical	C₂×#3500	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×D₄ ◁ C₂²×H(5,𝔽₂) ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

3434 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=i,e^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,e^b=ei,g^b=ghi,d^c=di,g^e=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72261611, #72261616, #72261685, …
maximal normal subgroup of	#72261611, #72261616, #72261685, …
maximal quotient of	#72313498

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400630);