#120245

order	512
nilpotency class	3
isoclinism	isoclinic class of #98904
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#380
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

570 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 28 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 27441);