#414376

order	512
nilpotency class	3
isoclinism	isoclinic class of #412862
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2858
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15170
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

947 conjugacy classes of subgroups; only the 192 normal subgroups are shown.

normal subgroups only

This group has 947 conjugacy classes of subgroups — too many to draw. Only its 192 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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

Polycyclic presentation:

2^9|a^2=ef,b^2=f,c^2=e,d^2=gh,g^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^c=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, 321572);