#7629786

order	512
nilpotency class	2
isoclinism	isoclinic class of D₄×#120
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂⁵
commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

8182 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 20 points:

(5,6)(7,10)(8,9)(11,12)(13,14)(15,18)(16,19)(17,20);(1,2)(3,4)(6,9)(10,12)(13,15,17,16)(14,18,20,19);(2,4)(13,15,17,16)(14,19,20,18);(1,2)(3,4)(13,15,17,16)(14,18,20,19);(2,4)(13,16,17,15)(14,19,20,18);(5,7,8,11)(6,10,9,12)

Polycyclic presentation:

2^9|b^2=h,c^2=h,d^2=h,e^2=h,f^2=g,b^a=bg,c^a=ch,c^b=ci,e^b=ei,d^c=di,e^d=ei

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308109, C₂×#7629786
maximal normal subgroup of	#72308109, C₂×#7629786
maximal quotient of	C₂×#7629786

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7536982);