#367419

order	512
nilpotency class	3
isoclinism	isoclinic class of #24805
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

844 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=egh,c^2=ef,d^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,d^b=di,g^c=gi

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, 274615);