#7631602

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×#546	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

9444 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,b^a=bg,c^a=ch,c^b=ci,f^b=fi,e^d=ei

Conjugacy classes & character table

122 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7538798);