#7689768

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	#36099	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7120 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

113 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7596964);