#343133

order	512
nilpotency class	3
isoclinism	isoclinic class of #343112
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2679
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3147	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

639 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 639 conjugacy classes of subgroups — too many to draw. Only its 122 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) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=efi,b^2=h,c^2=hi,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=defh,c^b=cg,d^b=de,e^b=eh,g^b=ghi,f^c=fi,g^c=ghi,e^d=eh,f^d=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 250329);