#7633648

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	#2675	C₂²
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)(3,7)(4,6)(5,8)(9,10)(11,12)(13,15)(14,16);(1,3,5,4)(2,6,8,7)(18,21)(22,24);(1,3,5,4)(2,7,8,6)(9,11)(10,14)(12,15)(13,16)(17,18)(19,22)(20,21)(23,24);(10,15)(11,16)(18,21)(22,24);(1,3,5,4)(2,7,8,6)(17,19,20,23)(18,22,21,24);(1,4,5,3)(2,6,8,7)(9,12,13,14)(10,11,15,16)

Polycyclic presentation:

2^9|b^2=g,c^2=g,e^2=gi,f^2=gh,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, 7540844);