#152299

order	512
nilpotency class	5
isoclinism	isoclinic class of #151054
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#367
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	C₂⁴	SD₃₂
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂³×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

249 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 249 conjugacy classes of subgroups — too many to draw. Only its 100 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 — 3 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dgh,d^2=ghi,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gh,h^a=hi,f^b=fi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319455, #72319634, D₄×#1426, …
maximal normal subgroup of	#72319455, #72319634, D₄×#1426

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 59495);