#7631046

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	#35001	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7590 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|b^2=gi,c^2=h,d^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, 7538242);