#7631858

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	#35088	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

8042 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)(5,6,8,10)(7,9,11,12)(13,14)(15,22)(16,20)(17,21)(18,23)(19,25)(24,28)(26,27);(5,6)(7,9)(8,10)(11,12)(13,15)(14,19)(16,26)(17,24)(18,25)(20,28)(21,27)(22,23);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,16,18,17)(14,20,23,21)(15,24,25,26)(19,27,22,28);(5,7,8,11)(6,9,10,12)(13,17,18,16)(14,21,23,20)(15,24,25,26)(19,27,22,28);(13,15)(14,22)(16,24)(17,26)(18,25)(19,23)(20,28)(21,27);(13,14)(15,22)(16,20)(17,21)(18,23)(19,25)(24,28)(26,27)

Polycyclic presentation:

2^9|a^2=h,c^2=i,d^2=g,b^a=bg,c^a=ch,c^b=ci,f^b=fi,e^d=ei

Conjugacy classes & character table

122 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72302359, #72302931, #72302957
maximal normal subgroup of	#72302359, #72302931, #72302957

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7539054);