#343420

order	512
nilpotency class	3
isoclinism	isoclinic class of #333407
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#555	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

774 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 774 conjugacy classes of subgroups — too many to draw. Only its 146 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) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 36 points:

(5,6,11,14,10,7,8,12)(9,17,16,15,19,20,18,13)(22,27,30,29)(23,31,32,33);(1,2)(3,4)(5,7,10,6)(8,12,11,14)(9,17,19,20)(13,18,15,16)(21,22,26,30)(23,28,32,36)(24,29,25,27)(31,35,33,34);(2,4)(5,8,10,11)(6,12,7,14)(9,16,19,18)(13,17,15,20)(21,23,26,32)(22,28,30,36)(24,31,25,33)(27,34,29,35);(5,9,10,19)(6,13,7,15)(8,16,11,18)(12,17,14,20)(22,29)(24,25)(27,30)(28,35)(31,33)(34,36)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 250616);