#115380

order	512
nilpotency class	3
isoclinism	isoclinic class of #115377
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1438
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

963 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72316773, #72316776
maximal normal subgroup of	#72316773, #72316776

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 22576);