#499810

order	512
nilpotency class	4
isoclinism	isoclinic class of #499803
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×B(4,𝔽₂)
commutator, Frattini, cosocle	C₂²×D₄	C₂⁴
prime radical	#32890	C₂
LCS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

1924 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=i,b^a=be,c^a=cf,d^a=de,g^a=gi,h^a=hi,c^b=cg,d^b=di,f^b=fh,e^c=eh,f^d=fhi,f^e=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72303827, #72312236, #72312247, …
maximal normal subgroup of	#72303827, #72312236, #72312247, …
maximal quotient of	#72303868, #72304015

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 407006);