#7670674

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	#37672	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7456 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)(13,14)(15,21)(16,20)(17,22)(18,19)(23,27)(24,26)(25,28);(5,6,8,9)(7,10,11,12)(13,15)(14,17)(16,23)(18,22)(19,21)(20,25)(24,28)(26,27);(2,4)(14,19)(15,22)(16,24)(25,27);(1,2)(3,4)(13,14)(15,17)(16,20)(18,19)(21,22)(23,25)(24,26)(27,28);(5,7,8,11)(6,10,9,12)(13,16,18,24)(14,20,19,26)(15,23,22,28)(17,25,21,27);(13,17)(14,15)(16,25)(18,21)(19,22)(20,23)(24,27)(26,28)

Polycyclic presentation:

2^9|b^2=gi,e^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	#72305286
maximal normal subgroup of	#72305286
maximal quotient of	#72306263

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7577870);