#7667451

order	512
nilpotency class	2
isoclinism	isoclinic class of #42996
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

6159 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)(10,13)(14,15)(17,18)(19,21)(20,22)(23,24);(2,5)(6,8)(17,19)(18,21)(20,24)(22,23);(9,10,12,13)(11,15,16,14)(17,18)(19,23)(20,22)(21,24);(9,11,12,16)(10,14,13,15)(19,24)(21,23);(18,22)(21,23);(1,3,4,7)(2,6,5,8)

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72303644, #72304032
maximal normal subgroup of	#72303644, #72304032
maximal quotient of	#72303672, #72303683, #72304064

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7574647);