#7629579

order	512
nilpotency class	2
isoclinism	isoclinic class of #35680
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	#2433	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5210 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

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, 7536775);