#7657428

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

8360 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)(5,6,9,12)(7,8,11,10)(13,14,20,19)(15,24,22,17)(16,21,28,27)(18,23,26,25);(1,2)(3,4)(13,15)(14,17)(16,23)(18,21)(19,24)(20,22)(25,28)(26,27);(5,7)(6,10)(8,12)(9,11)(13,16)(14,21)(15,25)(17,26)(18,24)(19,27)(20,28)(22,23);(13,17,20,24)(14,22,19,15)(16,26,28,18)(21,23,27,25);(5,7)(6,8)(9,11)(10,12)(13,18,20,26)(14,23,19,25)(15,21,22,27)(16,24,28,17);(5,8)(6,11)(7,12)(9,10)(13,19,20,14)(15,24,22,17)(16,27,28,21)(18,23,26,25)

Polycyclic presentation:

2^9|a^2=gh,d^2=i,e^2=i,f^2=i,b^a=bg,c^a=ch,d^a=di,c^b=ci,f^e=fhi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72238145, #72238172, #72238232
maximal normal subgroup of	#72238145, #72238172, #72238232
maximal quotient of	#72263569

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7564624);