#355099

order	512
nilpotency class	3
isoclinism	isoclinic class of #21776
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂³	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1924 conjugacy classes of subgroups; only the 394 normal subgroups are shown.

normal subgroups only

This group has 1924 conjugacy classes of subgroups — too many to draw. Only its 394 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=ce,f^b=fi,g^b=gi,g^d=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 262295);