#7670005

order	512
nilpotency class	2
isoclinism	isoclinic class of #42996
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

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

5310 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 32 points:

(1,2,4,5)(3,6,7,8)(9,10,12,13)(11,15,16,14)(17,18,22,26)(19,24,20,23)(21,29,30,25)(27,31,28,32);(2,5)(6,8)(10,13)(14,15)(17,19)(18,23)(20,22)(21,27)(24,26)(25,31)(28,30)(29,32);(9,11)(10,14)(12,16)(13,15)(17,20,22,19)(18,24,26,23)(21,27,30,28)(25,31,29,32);(11,16)(14,15)(19,20)(23,24)(27,28)(31,32);(17,21,22,30)(18,25,26,29)(19,27,20,28)(23,31,24,32);(1,3,4,7)(2,6,5,8)

Polycyclic presentation:

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

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, 7577201);