#7670013

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

5239 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=i,d^2=g,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, 7577209);