#7672776

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	C₂²×#120	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7510 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7579972);