#7641978

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	C₂×#3444	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

8202 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)(8,12)(10,11)(15,21)(18,25)(22,27)(26,28);(1,2)(3,4)(6,7)(10,11)(13,14,17,20)(15,18,21,25)(16,23,24,19)(22,28,27,26);(5,6,9,7)(8,10,12,11)(13,15)(14,18)(16,22)(17,21)(19,26)(20,25)(23,28)(24,27);(13,15)(14,18)(16,22)(17,21)(19,26)(20,25)(23,28)(24,27);(5,7,9,6)(8,11,12,10)(13,16,17,24)(14,19,20,23)(15,22,21,27)(18,26,25,28);(5,8)(6,10)(7,11)(9,12)(13,15)(14,18)(16,22)(17,21)(19,26)(20,25)(23,28)(24,27)

Polycyclic presentation:

2^9|b^2=h,c^2=i,e^2=hi,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^b=ehi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72304685, #72304797, #72311785, …
maximal normal subgroup of	#72304685, #72304797, #72311785, …
maximal quotient of	#72305085

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7549174);