#7636712

order	512
nilpotency class	2
isoclinism	isoclinic class of #37644
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

7798 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)(15,19)(16,18)(17,20);(1,2,4,3)(6,9)(10,12)(15,20)(17,19);(2,3)(13,15,16,20)(14,17,18,19);(14,18)(17,19);(1,3,4,2);(5,7,8,11)(6,10,9,12)

Polycyclic presentation:

2^9|b^2=hi,c^2=h,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dh,c^b=ci,e^c=ehi

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72282043, #72303354, #72308109, …
maximal normal subgroup of	#72282043, #72303354, #72308109, …
maximal quotient of	C₂×#7636712

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7543908);