#7626002

order	512
nilpotency class	2
isoclinism	isoclinic class of #3532
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂⁵
commutator	C₂²	C₂⁵×C₄
Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	C₂×#3444	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6914 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 36 points:

(1,2,3,4)(5,6)(7,13)(8,12)(9,11)(10,14)(15,19)(16,18)(17,20)(21,22)(23,26)(24,28)(25,27)(29,32)(30,33)(31,34)(35,36);(5,7)(6,10)(8,15)(9,14)(11,13)(12,17)(16,20)(18,19)(21,23)(22,26)(24,30)(25,29)(27,32)(28,33)(31,35)(34,36);(7,14)(10,13)(15,20)(17,19)(22,27)(26,32)(28,34)(33,36);(6,11)(10,13)(12,18)(17,19)(23,29)(26,32)(30,35)(33,36);(5,8,9,16)(6,12,11,18)(7,15,14,20)(10,17,13,19)(22,27)(26,32)(28,34)(33,36);(21,24,25,31)(22,28,27,34)(23,30,29,35)(26,33,32,36)

Polycyclic presentation:

2^9|a^2=i,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=cg,d^b=dh

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72314181, #72314184
maximal normal subgroup of	#72314181, #72314184

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7533198);