#7632017

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

8556 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)(6,9)(10,12)(13,14,19,17)(15,16,21,20)(18,22,27,25)(23,24,26,28);(1,2)(3,4)(13,15)(14,20)(16,17)(18,26)(19,21)(22,24)(23,27)(25,28);(5,6,8,9)(7,10,11,12)(13,16)(14,15)(17,21)(18,24)(19,20)(22,23)(25,26)(27,28);(5,7,8,11)(6,10,9,12)(13,17,19,14)(15,16,21,20)(18,25,27,22)(23,24,26,28);(13,16)(14,21)(15,17)(18,24)(19,20)(22,26)(23,25)(27,28);(13,18)(14,22)(15,23)(16,24)(17,25)(19,27)(20,28)(21,26)

Polycyclic presentation:

2^9|a^2=g,c^2=hi,d^2=h,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	#72262026, #72262117
maximal normal subgroup of	#72262026, #72262117

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7539213);