#7627223

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×#547	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

8386 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)(7,11)(8,9)(10,12)(13,14)(15,22)(16,24)(17,20)(18,21)(19,26)(23,25)(27,28);(6,9)(10,11)(13,15,18,20)(14,17,21,22)(16,23,26,27)(19,25,24,28);(5,6)(7,10)(8,9)(11,12)(13,15,18,20)(14,17,21,22)(16,23,26,27)(19,25,24,28);(5,7)(6,10)(8,12)(9,11)(13,16)(14,19)(15,23)(17,25)(18,26)(20,27)(21,24)(22,28);(13,14)(15,17)(16,19)(18,21)(20,22)(23,25)(24,26)(27,28);(13,17)(14,20)(15,21)(16,25)(18,22)(19,27)(23,24)(26,28)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=h,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.

Relationships

maximal subgroup of	#72304538, #72304562, #72304590, …
maximal normal subgroup of	#72304538, #72304562, #72304590, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7534419);