#7631252

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	C₂×#3511	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7342 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)(6,9)(10,12)(13,14,19,16)(15,22,21,18)(17,20,27,24)(23,28,25,26);(5,6)(7,10)(8,9)(11,12)(13,15)(14,18)(16,22)(17,25)(19,21)(20,28)(23,27)(24,26);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(14,16)(15,21)(20,24)(23,25);(5,7,8,11)(6,10,9,12)(13,16,19,14)(15,22,21,18)(17,20,27,24)(23,26,25,28);(13,17)(14,20)(15,23)(16,24)(18,26)(19,27)(21,25)(22,28);(13,18,19,22)(14,21,16,15)(17,26,27,28)(20,25,24,23)

Polycyclic presentation:

2^9|a^2=i,c^2=gh,d^2=g,f^2=i,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, 7538448);