#7627221

order	512
nilpotency class	2
isoclinism	isoclinic class of #63011
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂³	C₂⁶
commutator	C₂²	C₂⁵×C₄
prime radical	C₂²×#127	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7382 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)(7,10)(8,11)(9,12)(13,14,19,18)(15,22,17,20)(16,24,26,21)(23,27,25,28);(5,6)(7,10)(8,11)(9,12)(13,15,19,17)(14,20,18,22)(16,23,26,25)(21,27,24,28);(5,7)(6,9)(8,12)(10,11)(13,15,19,17)(14,20,18,22)(16,23,26,25)(21,27,24,28);(13,16)(14,21)(15,23)(17,25)(18,24)(19,26)(20,27)(22,28);(5,7)(6,10)(8,12)(9,11)(13,17,19,15)(14,20,18,22)(16,25,26,23)(21,27,24,28);(5,6)(7,9)(8,11)(10,12)(13,18,19,14)(15,22,17,20)(16,24,26,21)(23,28,25,27)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,c^b=cgh,f^e=fh

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7534417);