#7630676

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

9070 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 28 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72262098, #72262195, #72267580, …
maximal normal subgroup of	#72262098, #72262195, #72267580, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7537872);