#7678729

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

6846 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7585925);