#7651228

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²×C₄

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

8339 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)(6,9)(7,11);(1,2)(3,4)(6,9)(10,12)(13,14,18,19)(15,20,24,27)(16,17,21,26)(22,28,25,23);(5,6,8,9)(7,10,11,12)(14,19)(17,26)(20,27)(23,28);(5,7)(6,10)(8,11)(9,12)(13,15)(14,20)(16,25)(17,23)(18,24)(19,27)(21,22)(26,28);(13,16,18,21)(14,17,19,26)(15,22,24,25)(20,28,27,23);(13,17)(14,21)(15,23)(16,19)(18,26)(20,22)(24,28)(25,27)

Polycyclic presentation:

2^9|b^2=hi,c^2=h,e^2=hi,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^c=fhi,e^d=ehi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72302332, #72302996, #72303004
maximal normal subgroup of	#72302332, #72302996, #72303004

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7558424);