#7632016

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	#35346	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

8834 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)(6,9)(10,12)(13,14)(15,16)(17,20)(18,21)(19,23)(22,24)(25,26)(27,28);(1,2)(3,4)(13,14)(15,16)(17,25)(18,21)(19,23)(20,26)(22,27)(24,28);(5,6,8,9)(7,10,11,12)(13,15)(14,19)(16,21)(17,22)(18,23)(20,27)(24,25)(26,28);(5,7,8,11)(6,10,9,12)(13,16,18,19)(14,15,21,23)(17,24,26,27)(20,22,25,28);(13,15)(14,16)(17,22)(18,23)(19,21)(20,24)(25,27)(26,28);(13,17)(14,20)(15,22)(16,24)(18,26)(19,27)(21,25)(23,28)

Polycyclic presentation:

2^9|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	#72331266
maximal normal subgroup of	#72331266
maximal quotient of	#72331301, #72332172, #72332390

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7539212);