#7670685

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

7295 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)(5,6,9,8)(7,11,12,10)(13,14,19,17)(15,22,21,18)(16,20,24,25)(23,28,27,26);(5,7,9,12)(6,10,8,11)(13,15)(14,18)(16,23)(17,22)(19,21)(20,26)(24,27)(25,28);(2,4)(14,17)(15,21)(16,24)(26,28);(1,2)(3,4)(13,16)(14,20)(15,23)(17,25)(18,26)(19,24)(21,27)(22,28);(5,6,9,8)(7,10,12,11)(13,17,19,14)(15,22,21,18)(16,25,24,20)(23,28,27,26);(5,8,9,6)(7,11,12,10)(13,18,19,22)(14,21,17,15)(16,26,24,28)(20,27,25,23)

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72305243, #72305286
maximal normal subgroup of	#72305243, #72305286
maximal quotient of	#72305662, #72305739

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7577881);