#7631047

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

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

7322 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=h,d^2=g,e^2=g,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^b=ei,d^c=di,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, 7538243);