#122189

order	512
nilpotency class	3
isoclinism	isoclinic class of #122187
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1446
commutator, prime radical	C₂⁵	C₂²×C₄
Frattini, cosocle	C₂⁴×C₄	C₂³
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

749 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 29385);