#455198

order	512
nilpotency class	3
isoclinism	isoclinic class of #455198
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2620
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16261
prime radical	#60469	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1250 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:

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

Polycyclic presentation:

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

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, 362394);