#343092

order	512
nilpotency class	3
isoclinism	isoclinic class of #343075
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	#3245	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

690 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 690 conjugacy classes of subgroups — too many to draw. Only its 128 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=ef,b^2=hi,c^2=hi,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=defhi,c^b=cg,d^b=de,e^b=eh,g^b=gh,f^c=fi,g^c=gh,e^d=eh,f^d=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 250288);