#233552

order	512
nilpotency class	3
isoclinism	isoclinic class of #233549
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	C₂⁵	C₂⁴
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

2280 conjugacy classes of subgroups; only the 198 normal subgroups are shown.

normal subgroups only

This group has 2280 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)(11,15,21,29)(12,16,23,32)(13,24,26,17)(19,27,35,30)(20,33,25,28)(22,36,38,31)(34,40,37,39);(1,3)(2,5)(4,7)(6,8)(9,11,14,21)(10,15,18,29)(12,19,23,35)(13,25,26,20)(16,30,32,27)(17,33,24,28)(22,37,38,34)(31,39,36,40);(5,8)(9,12)(10,16)(11,19)(13,22)(14,23)(15,27)(17,31)(18,32)(20,34)(21,35)(24,36)(25,37)(26,38)(28,39)(29,30)(33,40);(9,13,14,26)(10,17,18,24)(11,20,21,25)(12,22,23,38)(15,28,29,33)(16,31,32,36)(19,34,35,37)(27,39,30,40)

Polycyclic presentation:

2^9|a^2=h,b^2=h,d^2=h,b^a=be,c^a=cf,d^a=dh,g^a=ghi,c^b=cg,d^b=dh,f^b=fi,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, 140748);