#120845

order	512
nilpotency class	3
isoclinism	isoclinic class of #120844
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1442
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂⁵	#41
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

545 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)(3,9)(4,12,14,7)(5,8)(6,11)(10,15,13,16)(17,18,23,19,21,25,22,27)(20,26,31,32,29,28,30,24);(1,3)(2,6,8,9)(4,13)(5,11)(7,16,12,15)(10,14)(17,19,22,25,21,27,23,18)(20,28,30,24,29,26,31,32);(1,4)(2,7)(3,10,11,13)(5,14)(6,15,9,16)(8,12)(17,20,21,29)(18,24)(19,26)(22,30,23,31)(25,32)(27,28)

Polycyclic presentation:

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

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, 28041);