#7628646

order	512
nilpotency class	2
isoclinism	isoclinic class of #34980
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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7776 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

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, 7535842);