#470948

order	512
nilpotency class	3
isoclinism	isoclinic class of #457608
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2735
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16720
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

695 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 378144);