#7631589

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

8560 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)(5,6,10,9)(7,12,11,8)(13,14)(15,17)(16,23)(18,20)(19,21)(22,25)(24,26)(27,28);(5,7)(6,8)(9,12)(10,11)(13,15)(14,16)(17,21)(18,26)(19,23)(20,25)(22,27)(24,28);(1,2)(3,4)(5,8,10,12)(6,11,9,7)(13,16,19,17)(14,15,21,23)(18,24,27,25)(20,22,28,26);(5,9,10,6)(7,12,11,8)(13,17,19,16)(14,15,21,23)(18,25,27,24)(20,22,28,26);(13,15)(14,17)(16,21)(18,22)(19,23)(20,25)(24,28)(26,27);(13,18)(14,20)(15,22)(16,24)(17,25)(19,27)(21,28)(23,26)

Polycyclic presentation:

2^9|a^2=gi,c^2=g,d^2=g,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.

Relationships

maximal subgroup of	#72262059, #72262090
maximal normal subgroup of	#72262059, #72262090

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7538785);