#122041

order	512
nilpotency class	3
isoclinism	isoclinic class of #10390
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#393
commutator, prime radical	C₂⁵	C₂²×C₄
Frattini, cosocle	C₂⁴×C₄	C₂³
socle	C₂²	#1446
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ 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)(11,19,22,35,21,15,27,29)(12,16,24,32,23,31,25,34)(20,30,37,40,36,28,38,39);(1,3)(2,5)(4,8)(6,7)(9,11)(10,15,17,19)(12,20)(13,22)(14,21)(16,30,31,28)(18,29,33,35)(23,36)(24,37)(25,38)(26,27)(32,40,34,39);(1,2)(3,5)(4,6)(7,8)(9,12)(10,16,17,31)(11,20)(13,25)(14,23)(15,28,19,30)(18,34,33,32)(21,36)(22,38)(24,26)(27,37)(29,40,35,39)

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,e^a=ei,f^a=fi,c^b=cf,d^b=di,e^b=eh,g^b=gh,d^c=dhi,e^c=ei,g^c=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72303820
maximal normal subgroup of	#72303820
maximal quotient of	#72303932

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 29237);