#7720314

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

7541 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)(6,10)(8,11)(13,14)(15,21)(16,24)(17,19)(18,22)(20,25)(23,26)(27,28);(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,15,17,22)(14,18,19,21)(16,23,25,28)(20,26,24,27);(6,10)(7,12)(14,19)(15,22)(20,24)(23,28);(13,14)(15,18)(16,20)(17,19)(21,22)(23,26)(24,25)(27,28);(5,7)(6,11)(8,10)(9,12)(13,15)(14,18)(16,23)(17,22)(19,21)(20,26)(24,27)(25,28);(5,8,9,11)(6,7,10,12)(13,16,17,25)(14,20,19,24)(15,23,22,28)(18,26,21,27)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72236855, #72238194, #72238229
maximal normal subgroup of	#72236855, #72238194, #72238229
maximal quotient of	#72263404

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7627510);