#457591

order	512
nilpotency class	3
isoclinism	isoclinic class of #454109
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2670
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16297
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1542 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)(3,12,15,7)(4,8)(5,21)(6,10)(9,19)(11,22)(13,23,30,29)(14,31,32,26)(16,24,17,25)(18,28)(20,27);(1,3)(2,7)(4,14)(5,17)(6,15)(8,26)(9,23)(10,12)(11,25)(13,18)(16,22)(19,30)(20,32)(21,24)(27,31)(28,29);(1,4)(2,8)(3,13,15,30)(5,18)(6,20)(7,23,12,29)(9,11)(10,27)(14,17,32,16)(19,22)(21,28)(24,26,25,31);(1,4)(2,9)(3,14)(5,18)(6,20)(7,23)(8,21)(10,28)(11,27)(12,29)(13,17)(15,32)(16,30)(19,22)(24,26)(25,31)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318583, #72318935, #72318940, …
maximal normal subgroup of	#72318583, #72318935, #72318940, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 364787);