#234367

order	512
nilpotency class	3
isoclinism	isoclinic class of #234365
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2234 conjugacy classes of subgroups; only the 198 normal subgroups are shown.

normal subgroups only

This group has 2234 conjugacy classes of subgroups — too many to draw. Only its 198 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) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2)(3,5)(4,6)(7,8)(9,10,14,18)(11,19,22,34)(12,16,23,32)(13,25,28,17)(15,26,30,21)(20,36,37,40)(24,35,27,33)(29,38,31,39);(1,3)(2,5)(4,7)(6,8)(9,11,14,22)(10,15,18,30)(12,20,23,37)(13,26,28,21)(16,31,32,29)(17,34,25,19)(24,39,27,38)(33,40,35,36);(5,8)(9,12)(10,16)(11,20)(13,27)(14,23)(15,29)(17,35)(18,32)(19,36)(21,39)(22,37)(24,28)(25,33)(26,38)(30,31)(34,40);(12,23)(16,32)(20,37)(24,27)(29,31)(33,35)(36,40)(38,39)

Polycyclic presentation:

2^9|a^2=h,b^2=h,e^2=h,b^a=be,c^a=cf,e^a=eh,g^a=ghi,c^b=cg,e^b=eh,f^b=fi,d^c=dh,e^c=eh

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, 141563);