#7638041

order	512
nilpotency class	2
isoclinism	isoclinic class of #37644
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

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

5735 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 24 points:

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

Polycyclic presentation:

2^9|c^2=h,d^2=hi,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	#72282973, #72304359, #72304441, …
maximal normal subgroup of	#72282973, #72304359, #72304441, …
maximal quotient of	C₂×#7638041

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7545237);