#7735707

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

8482 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)(7,9)(8,12)(10,11)(13,14)(15,22)(16,19)(17,20)(18,21)(23,28)(24,26)(25,27);(1,2,3,4)(5,7,10,8)(6,9,11,12)(13,15,18,16)(14,19,21,22)(17,23,25,24)(20,26,27,28);(6,11)(7,8)(15,16)(17,25)(19,22)(20,27);(13,16,18,15)(14,19,21,22)(17,24,25,23)(20,26,27,28);(5,8,10,7)(6,9,11,12)(13,17)(14,20)(15,23)(16,24)(18,25)(19,26)(21,27)(22,28);(5,9)(6,7)(8,11)(10,12)(13,14)(15,22)(16,19)(17,20)(18,21)(23,28)(24,26)(25,27)

Polycyclic presentation:

2^9|b^2=gh,d^2=hi,e^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,e^c=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72304549, #72304592, #72304611
maximal normal subgroup of	#72304549, #72304592, #72304611
maximal quotient of	#72267287, #72267303

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7642903);