#395528

order	512
nilpotency class	3
isoclinism	isoclinic class of #394328
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2617
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14725
prime radical	#60387	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2122 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72215931, #72219688, #72219689, …
maximal normal subgroup of	#72215931, #72219688, #72219689, …
maximal quotient of	C₂×#395528

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 302724);