#7672137

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	C₂⁵	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6940 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)(5,6,8,10)(7,11,12,9)(13,14)(15,18)(16,24)(17,20)(19,25)(21,23)(22,27)(26,28);(5,7,8,12)(6,9,10,11)(13,14)(15,18)(16,19)(17,20)(21,23)(22,26)(24,25)(27,28);(2,4)(13,14,17,20)(15,21,23,18)(16,19,25,24)(22,27,28,26);(1,2)(3,4)(13,14)(15,18)(16,19)(17,20)(21,23)(22,26)(24,25)(27,28);(13,15,17,23)(14,18,20,21)(16,22,25,28)(19,26,24,27);(13,16)(14,19)(15,22)(17,25)(18,26)(20,24)(21,27)(23,28)

Polycyclic presentation:

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

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, 7579333);