#7631583

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	C₂×#3485	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

8744 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)(6,9)(7,11)(13,14,18,23)(15,19,25,26)(16,21,27,20)(17,22,28,24);(5,6)(7,10)(8,9)(11,12)(13,15)(14,19)(16,22)(17,20)(18,25)(21,28)(23,26)(24,27);(1,2)(3,4)(5,7,8,11)(6,10,9,12)(13,16)(14,20)(15,24)(17,26)(18,27)(19,28)(21,23)(22,25);(13,16)(14,21)(15,22)(17,26)(18,27)(19,28)(20,23)(24,25);(13,15)(14,19)(16,24)(17,21)(18,25)(20,28)(22,27)(23,26);(13,17)(14,22)(15,21)(16,26)(18,28)(19,27)(20,25)(23,24)

Polycyclic presentation:

2^9|a^2=i,c^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	#72256684, #72256693, #72256695
maximal normal subgroup of	#72256684, #72256693, #72256695

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7538779);