#7670668

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
prime radical	#35000	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7342 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)(6,9)(10,12)(14,19)(17,25)(20,26)(22,28);(5,6,8,9)(7,10,11,12)(13,14)(15,20)(16,17)(18,19)(21,22)(23,26)(24,25)(27,28);(2,4)(14,19)(16,24)(20,26)(21,27);(1,2)(3,4)(5,7,8,11)(6,10,9,12)(13,15,18,23)(14,20,19,26)(16,21,24,27)(17,22,25,28);(13,16)(14,17)(15,21)(18,24)(19,25)(20,22)(23,27)(26,28);(13,17)(14,16)(15,22)(18,25)(19,24)(20,21)(23,28)(26,27)

Polycyclic presentation:

2^9|b^2=gi,d^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	#72305257, #72305265, #72305286
maximal normal subgroup of	#72305257, #72305265, #72305286
maximal quotient of	#72306173

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7577864);