#120244

order	512
nilpotency class	3
isoclinism	isoclinic class of #98904
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

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	C₂²⋊Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

544 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=di,c^2=d,d^2=h,e^2=hi,g^2=h,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, 27440);