#7628462

order	512
nilpotency class	2
isoclinism	isoclinic class of #34980
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂⁵
commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#35166	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7338 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 28 points:

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

Polycyclic presentation:

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

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, 7535658);