#7683256

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

8279 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)(7,11)(8,9)(10,12)(13,14)(15,20)(16,22)(17,18)(19,21)(23,27)(24,26)(25,28);(5,7)(6,10)(8,11)(9,12)(13,15,19,24)(14,20,21,26)(16,23,25,17)(18,22,27,28);(2,4)(5,8,9,6)(7,11,12,10)(14,21)(15,24)(16,25)(18,27);(1,2)(3,4)(5,7)(6,11)(8,10)(9,12)(13,16)(14,22)(15,23)(17,24)(18,26)(19,25)(20,27)(21,28);(13,17,19,23)(14,18,21,27)(15,25,24,16)(20,28,26,22);(13,18)(14,23)(15,22)(16,26)(17,21)(19,27)(20,25)(24,28)

Polycyclic presentation:

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

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72302317, #72303004, #72303006
maximal normal subgroup of	#72302317, #72303004, #72303006

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7590452);