#395628

order	512
nilpotency class	3
isoclinism	isoclinic class of #395628
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14725
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3021	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2148 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)(3,13)(4,19,21,25)(5,11)(6,10)(7,12)(8,16)(9,20,26,22)(14,18)(15,23,30,32)(17,24)(27,29,28,31);(1,3,5,16,7,18,6,17)(2,8,10,13,12,24,11,14)(4,15,20,29,21,30,22,31)(9,23,25,27,26,32,19,28);(1,2)(3,14)(4,9)(5,10)(6,11)(7,12)(8,16)(13,18)(15,28)(17,24)(19,22)(20,25)(21,26)(23,29)(27,30)(31,32);(1,4)(2,9)(3,15)(5,20)(6,22)(7,21)(8,23)(10,25)(11,19)(12,26)(13,27)(14,28)(16,29)(17,31)(18,30)(24,32)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72237064, #72237067, #72237074, …
maximal normal subgroup of	#72237064, #72237067, #72237074, …
maximal quotient of	C₂×#395628

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 302824);