#7669769

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	#2428	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4691 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)(10,13)(14,16)(17,18,21,24)(19,22,25,29)(20,27,28,23)(26,32,31,30);(1,3,5,4)(2,6,8,7)(17,19)(18,22)(20,26)(21,25)(23,30)(24,29)(27,32)(28,31);(1,3,5,4)(2,7,8,6)(9,10)(11,14)(12,13)(15,16)(19,25)(20,28)(22,29)(23,27);(1,3,5,4)(2,7,8,6)(10,13)(14,16)(19,25)(22,29)(26,31)(30,32);(17,20,21,28)(18,23,24,27)(19,26,25,31)(22,30,29,32);(1,4,5,3)(2,6,8,7)(9,11,12,15)(10,14,13,16)

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=g,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, 7576965);