#7670678

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

7189 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)(5,6,8,10)(7,9,11,12)(16,24)(19,26)(22,28)(25,27);(5,6)(7,9)(8,10)(11,12)(13,14,17,20)(15,18,23,21)(16,19,24,26)(22,25,28,27);(2,4)(13,14)(15,21)(16,19)(17,20)(18,23)(22,27)(24,26)(25,28);(1,2)(3,4);(13,15)(14,18)(16,22)(17,23)(19,25)(20,21)(24,28)(26,27);(5,7,8,11)(6,9,10,12)(13,16)(14,19)(15,22)(17,24)(18,25)(20,26)(21,27)(23,28)

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72305276, #72305286
maximal normal subgroup of	#72305276, #72305286
maximal quotient of	#72306138, #72306150

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7577874);