#7641989

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	C₂²×#532	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

8376 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)(13,14,18,17)(15,21,23,19)(16,24,25,20)(22,26,28,27);(1,2)(3,4)(6,7)(10,11)(14,17)(19,21)(20,24)(26,27);(5,6,9,7)(8,10,12,11)(13,15)(14,19)(16,22)(17,21)(18,23)(20,26)(24,27)(25,28);(13,16,18,25)(14,20,17,24)(15,22,23,28)(19,26,21,27);(5,7,9,6)(8,11,12,10)(13,17,18,14)(15,21,23,19)(16,24,25,20)(22,27,28,26);(5,8)(6,10)(7,11)(9,12)(13,15)(14,19)(16,22)(17,21)(18,23)(20,26)(24,27)(25,28)

Polycyclic presentation:

2^9|a^2=h,c^2=i,d^2=h,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	#72304728, #72304739, #72304797, …
maximal normal subgroup of	#72304728, #72304739, #72304797, …
maximal quotient of	#72305000, #72305142

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7549185);