#7733771

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	#34957	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

8034 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)(13,14,18,22)(15,23,24,19)(16,26,27,20)(17,21,28,25);(1,2,3,4)(5,6)(7,11)(8,9)(10,12)(15,24)(17,28)(19,23)(21,25);(6,9)(10,11)(13,15,18,24)(14,19,22,23)(16,21,27,25)(17,20,28,26);(5,6)(7,10)(8,9)(11,12);(5,6)(7,10)(8,9)(11,12)(13,16,18,27)(14,20,22,26)(15,25,24,21)(17,23,28,19);(5,7)(6,10)(8,12)(9,11)(13,17)(14,21)(15,20)(16,23)(18,28)(19,27)(22,25)(24,26)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^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	#72304593, #72304601, #72304636
maximal normal subgroup of	#72304593, #72304601, #72304636

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7640967);