#7632830

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	#35512	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

8283 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:

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

Polycyclic presentation:

2^9|c^2=gh,d^2=gh,f^2=i,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, 7540026);