#119727

order	512
nilpotency class	3
isoclinism	isoclinic class of #119726
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1440
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

846 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 846 conjugacy classes of subgroups — too many to draw. Only its 126 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) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=d,e^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fi,c^b=cf,e^b=ei,g^b=gh,e^c=ehi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72267645
maximal normal subgroup of	#72267645
maximal quotient of	#72267649

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 26923);