#454663

order	512
nilpotency class	3
isoclinism	isoclinic class of #453995
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2812
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16255
prime radical	#60387	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1782 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72263012, #72267112, #72318806, …
maximal normal subgroup of	#72263012, #72267112, #72318806, …
maximal quotient of	#72262982

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 361859);