#454672

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	#16255
prime radical	#60387	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1783 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:

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

Polycyclic presentation:

2^9|a^2=ef,e^2=i,h^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,e^d=ei,f^d=fi,g^d=gi,h^d=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72263012, #72267112
maximal normal subgroup of	#72263012, #72267112
maximal quotient of	#72262974

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 361868);