#7633643

order	512
nilpotency class	2
isoclinism	isoclinic class of #2668
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄²	C₂⁴
commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂³ ▷ C₁

5748 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)(11,13)(12,14)(15,16)(17,18)(19,23)(20,22)(21,24);(2,5)(6,8)(19,24)(21,23);(9,11,13,10)(12,15,16,14)(17,19,20,24)(18,21,22,23);(10,11)(14,15)(18,22)(21,23);(9,12,13,16)(10,14,11,15);(1,3,4,7)(2,6,5,8)

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72304025, #72304317, #72304359, …
maximal normal subgroup of	#72304025, #72304317, #72304359, …
maximal quotient of	#72304035

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7540839);