#7633217

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

8930 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:

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

Polycyclic presentation:

2^9|a^2=gi,c^2=hi,f^2=hi,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	#72336516
maximal normal subgroup of	#72336516
maximal quotient of	#72336730

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7540413);