#404717

order	512
nilpotency class	3
isoclinism	isoclinic class of #397902
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

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	C₂×#2017	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1216 conjugacy classes of subgroups; only the 206 normal subgroups are shown.

normal subgroups only

This group has 1216 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) = 128

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

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

Polycyclic presentation:

2^9|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, 311913);