#99786

order	512
nilpotency class	3
isoclinism	isoclinic class of #96924
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

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

1328 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfgh,c^2=gh,d^2=gh,f^2=i,b^a=bd,c^a=ce,d^a=dg,c^b=cgi,d^b=dh,f^b=fh

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72279383, #72279399, #72279441
maximal normal subgroup of	#72279383, #72279399, #72279441
maximal quotient of	#72279816, #72280129

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6982);