#427878

order	512
nilpotency class	3
isoclinism	isoclinic class of #419464
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2557
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15375
prime radical	#23412	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1221 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)(4,18)(5,20)(6,24)(7,11)(8,15)(9,19)(10,25)(13,29)(14,26)(16,28)(17,27)(21,23)(22,31)(30,32);(1,3)(2,8)(4,13)(5,22)(6,16)(7,17)(9,29)(10,27)(11,28)(12,24)(14,23)(15,25)(18,32)(19,26)(20,30)(21,31);(1,4)(2,9)(3,13)(5,11)(6,20)(7,21)(8,22)(10,23)(12,14)(15,26)(16,30)(17,31)(18,24)(19,25)(27,32)(28,29);(1,5,7,9)(2,4,11,21)(3,14)(6,23,25,18)(8,26)(10,19,24,20)(12,13)(15,22)(16,29)(17,32)(27,31)(28,30)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 335074);