#7633645

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
prime radical	#52169	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂³ ▷ C₁

5042 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|b^2=h,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.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7540841);