#405987

order	512
nilpotency class	3
isoclinism	isoclinic class of #395586
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2411
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14922
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1010 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)(3,13,17,29)(4,9)(5,14,21,10)(6,11)(7,12)(8,16,24,32)(15,31)(18,25)(19,26)(20,30,23,27)(22,28);(1,3)(2,8,12,24)(4,14,19,10)(5,15)(6,16)(7,17)(9,20)(11,13,28,29)(18,30,31,27)(21,25)(22,32)(23,26);(1,4)(2,9)(3,14)(5,13)(6,18)(7,19)(8,23)(10,17)(11,25)(12,26)(15,28)(16,30)(20,24)(21,29)(22,31)(27,32);(1,5,6,20,7,21,22,23)(2,10,11,27,12,14,28,30)(3,15,16,9,17,25,32,26)(4,13,18,24,19,29,31,8)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 313183);