#7733973

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

8846 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,d^2=h,e^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,e^c=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72304562, #72304580, #72304593
maximal normal subgroup of	#72304562, #72304580, #72304593

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7641169);