#405069

order	512
nilpotency class	3
isoclinism	isoclinic class of #397902
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#127
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2397
prime radical	C₂⁵	#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

937 conjugacy classes of subgroups; only the 206 normal subgroups are shown.

normal subgroups only

This group has 937 conjugacy classes of subgroups — too many to draw. Only its 206 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=f,d^2=g,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,f^b=fi,e^c=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 312265);