#209482

order	512
nilpotency class	3
isoclinism	isoclinic class of #18414
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#1595	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1393 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=g,d^2=gi,e^2=h,b^a=be,c^a=cf,d^a=di,e^a=eh,e^b=eh,f^b=fi,d^c=dh,e^c=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 116678);