#7630504

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	C₂⁴×D₄	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

8754 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)(5,6,8,10)(7,11,12,9)(13,14,19,20)(15,22,16,17)(18,26,27,21)(23,25,24,28);(5,7,8,12)(6,9,10,11)(13,15,19,16)(14,17,20,22)(18,23,27,24)(21,25,26,28);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(14,20)(15,16)(21,26)(23,24);(13,16,19,15)(14,17,20,22)(18,24,27,23)(21,25,26,28);(13,17)(14,15)(16,20)(18,25)(19,22)(21,23)(24,26)(27,28);(13,18)(14,21)(15,23)(16,24)(17,25)(19,27)(20,26)(22,28)

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7537700);