#343025

order	512
nilpotency class	3
isoclinism	isoclinic class of #343003
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

678 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 678 conjugacy classes of subgroups — too many to draw. Only its 138 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=h,c^2=i,d^2=i,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=def,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

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 250221);