#7782729

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

8136 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)(8,9)(10,12)(13,14,18,23)(15,19,17,22)(16,21,26,20)(24,28,25,27);(1,2)(3,4)(13,15,18,17)(14,19,23,22)(16,25,26,24)(20,28,21,27);(6,9)(10,11)(13,16,18,26)(14,20,23,21)(15,24,17,25)(19,27,22,28);(5,6)(7,10)(8,9)(11,12)(13,16)(14,21)(15,25)(17,24)(18,26)(19,27)(20,23)(22,28);(5,7)(6,10)(8,12)(9,11)(13,14)(15,19)(16,20)(17,22)(18,23)(21,26)(24,27)(25,28);(13,17,18,15)(14,22,23,19)(16,24,26,25)(20,27,21,28)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,f^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,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	#72302981, #72303009
maximal normal subgroup of	#72302981, #72303009

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7689925);