#122194

order	512
nilpotency class	3
isoclinism	isoclinic class of #122191
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1446
commutator, prime radical	C₂⁵	C₂²×C₄
Frattini, cosocle	C₂⁴×C₄	C₂³
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 816 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) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=f,g^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fi,c^b=cf,d^b=di,e^b=eh,g^b=gh,d^c=dhi,e^c=ei

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, 29390);