#7816037

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

8336 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)(7,11)(8,10)(9,12)(13,14,19,22)(15,20,23,18)(16,17,21,27)(24,25,28,26);(1,2)(3,4)(13,15)(14,20)(16,24)(17,25)(18,22)(19,23)(21,28)(26,27);(5,6,8,10)(7,9,12,11)(15,23)(18,20)(24,28)(25,26);(5,6)(7,9)(8,10)(11,12)(13,16)(14,17)(15,24)(18,26)(19,21)(20,25)(22,27)(23,28);(5,7)(6,9)(8,12)(10,11)(13,17)(14,21)(15,25)(16,22)(18,24)(19,27)(20,28)(23,26);(13,18)(14,15)(16,26)(17,24)(19,20)(21,25)(22,23)(27,28)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72261968, #72262059, #72282531
maximal normal subgroup of	#72261968, #72262059, #72282531
maximal quotient of	#72283396, #72283887

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7723233);