#7672869

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	C₂²×#121	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7096 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)(13,14,19,21)(15,20,16,22)(17,18,27,28)(23,24,25,26);(5,7,8,12)(6,9,10,11)(13,15)(14,20)(16,19)(17,23)(18,24)(21,22)(25,27)(26,28);(2,3)(14,21)(15,16)(17,27)(24,26);(1,3,4,2)(13,16,19,15)(14,22,21,20)(17,25,27,23)(18,26,28,24);(13,17)(14,18)(15,23)(16,25)(19,27)(20,24)(21,28)(22,26);(13,18)(14,17)(15,24)(16,26)(19,28)(20,23)(21,27)(22,25)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7580065);