#7636721

order	512
nilpotency class	2
isoclinism	isoclinic class of #37644
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂⁵
commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#39340	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6962 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 28 points:

(5,6)(7,10)(8,9)(11,12)(13,14)(15,22)(16,25)(17,20)(18,21)(19,24)(23,27)(26,28);(1,2,4,3)(6,9)(10,12);(2,3)(13,15)(14,19)(16,23)(17,26)(18,24)(20,27)(21,22)(25,28);(13,16,18,17)(14,20,21,25)(15,23,24,26)(19,27,22,28);(1,3,4,2)(13,17,18,16)(14,20,21,25)(15,23,24,26)(19,28,22,27);(5,7,8,11)(6,10,9,12)

Polycyclic presentation:

2^9|b^2=i,d^2=h,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.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7543917);