#7670022

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
prime radical	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4669 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)(3,6)(4,5)(7,8)(11,15)(13,16)(17,18,22,20)(19,23,25,27)(21,28,29,24)(26,32,31,30);(2,5)(6,8)(9,10,12,14)(11,13,15,16)(17,19)(18,23)(20,27)(21,26)(22,25)(24,30)(28,32)(29,31);(9,11)(10,13)(12,15)(14,16)(17,20,22,18)(19,23,25,27)(21,24,29,28)(26,32,31,30);(11,15)(13,16)(19,25)(23,27)(26,31)(30,32);(17,21,22,29)(18,24,20,28)(19,26,25,31)(23,30,27,32);(1,3,4,7)(2,6,5,8)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=i,e^2=i,f^2=g,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, 7577218);