#396534

order	512
nilpotency class	3
isoclinism	isoclinic class of #393934
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂²×H(5,𝔽₂)	C₂²
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

2434 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=efi,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,h^b=hi,h^c=hi,e^d=ei,f^d=fi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72267126, #72267130, #72267136, …
maximal normal subgroup of	#72267126, #72267130, #72267136, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 303730);