#413050

order	512
nilpotency class	3
isoclinism	isoclinic class of #395151
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₈	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₄×#95 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1820 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=fi,c^2=e,d^2=h,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 320246);