#343098

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	C₂³	#534
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

654 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 654 conjugacy classes of subgroups — too many to draw. Only its 132 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:

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

Polycyclic presentation:

2^9|a^2=efi,b^2=h,c^2=hi,d^2=i,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, 250294);