#7670682

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

7351 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)(6,10)(11,12)(13,14)(15,23)(16,21)(17,27)(18,20)(19,24)(22,25)(26,28);(5,6,9,10)(7,11,8,12)(13,15)(14,19)(16,25)(17,26)(18,24)(20,23)(21,27)(22,28);(2,4)(14,20)(15,24)(16,26)(22,27);(1,2)(3,4)(5,7,9,8)(6,11,10,12)(13,16,18,26)(14,21,20,28)(15,25,24,17)(19,27,23,22);(13,14)(15,19)(16,21)(17,22)(18,20)(23,24)(25,27)(26,28);(5,8,9,7)(6,12,10,11)(13,17,18,25)(14,22,20,27)(15,26,24,16)(19,28,23,21)

Polycyclic presentation:

2^9|b^2=gi,d^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	#72305286
maximal normal subgroup of	#72305286
maximal quotient of	#72306252, #72306254

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7577878);