#343117

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
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 683 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)(17,18,23,24,21,26,22,27)(19,28,30,31,25,20,29,32);(1,2,5,9)(3,7,12,15)(4,8,13,6)(10,14,11,16)(17,19)(18,20)(21,25)(22,29)(23,30)(24,32)(26,28)(27,31);(1,3,5,12)(2,7,9,15)(4,11,13,10)(6,14,8,16)(18,24)(19,29)(22,23)(25,30)(26,27)(31,32);(2,8)(4,13)(6,9)(7,16)(10,11)(14,15)(17,20)(18,25)(19,26)(21,28)(22,32)(23,31)(24,30)(27,29)

Polycyclic presentation:

2^9|a^2=efh,b^2=h,c^2=h,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, 250313);